Finite Orbits: Returns, Least Periods and Transient States
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261013-advance-20261021-r03-orbit-fixed-periodic-foundations · 2026-10-13 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. One State’s Fixed-Point Condition
Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=-7, b=-9. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=0. Give (modulus,a,b,x,T(x),T(x)=x?).
View answer and explanation
(5, -7, -9, 0, 1, No)
Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, -7, -9, 0, 1, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate
2. One State’s Fixed-Point Condition
Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=-6, b=-7. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=1. Give (modulus,a,b,x,T(x),T(x)=x?).
View answer and explanation
(6, -6, -7, 1, 5, No)
Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, -6, -7, 1, 5, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate
3. One State’s Fixed-Point Condition
Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=-5, b=-5. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=2. Give (modulus,a,b,x,T(x),T(x)=x?).
View answer and explanation
(7, -5, -5, 2, 6, No)
Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, -5, -5, 2, 6, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate
4. One State’s Fixed-Point Condition
Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=-4, b=-3. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=3. Give (modulus,a,b,x,T(x),T(x)=x?).
View answer and explanation
(8, -4, -3, 3, 1, No)
Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, -4, -3, 3, 1, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate
5. One State’s Fixed-Point Condition
Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=-3, b=-1. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=4. Give (modulus,a,b,x,T(x),T(x)=x?).
View answer and explanation
(9, -3, -1, 4, 5, No)
Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, -3, -1, 4, 5, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate
6. One State’s Fixed-Point Condition
Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=-2, b=1. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=5. Give (modulus,a,b,x,T(x),T(x)=x?).
View answer and explanation
(10, -2, 1, 5, 1, No)
Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, -2, 1, 5, 1, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate
7. One State’s Fixed-Point Condition
Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=-1, b=3. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=1. Give (modulus,a,b,x,T(x),T(x)=x?).
View answer and explanation
(5, -1, 3, 1, 2, No)
Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, -1, 3, 1, 2, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate
8. One State’s Fixed-Point Condition
Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=0, b=5. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=1. Give (modulus,a,b,x,T(x),T(x)=x?).
View answer and explanation
(6, 0, 5, 1, 5, No)
Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, 0, 5, 1, 5, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate
9. One State’s Fixed-Point Condition
Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=1, b=7. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=1. Give (modulus,a,b,x,T(x),T(x)=x?).
View answer and explanation
(7, 1, 7, 1, 1, Yes)
Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, 1, 7, 1, 1, Yes).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate
10. One State’s Fixed-Point Condition
Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=2, b=9. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=1. Give (modulus,a,b,x,T(x),T(x)=x?).
View answer and explanation
(8, 2, 9, 1, 3, No)
Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, 2, 9, 1, 3, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate
11. One State’s Fixed-Point Condition
Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=3, b=11. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=1. Give (modulus,a,b,x,T(x),T(x)=x?).
View answer and explanation
(9, 3, 11, 1, 5, No)
Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, 3, 11, 1, 5, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate
12. One State’s Fixed-Point Condition
Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=4, b=13. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=1. Give (modulus,a,b,x,T(x),T(x)=x?).
View answer and explanation
(10, 4, 13, 1, 7, No)
Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, 4, 13, 1, 7, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate
13. One State’s Fixed-Point Condition
Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=5, b=15. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=2. Give (modulus,a,b,x,T(x),T(x)=x?).
View answer and explanation
(5, 5, 15, 2, 0, No)
Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 5, 15, 2, 0, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate
14. One State’s Fixed-Point Condition
Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=6, b=17. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=1. Give (modulus,a,b,x,T(x),T(x)=x?).
View answer and explanation
(6, 6, 17, 1, 5, No)
Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, 6, 17, 1, 5, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate
15. One State’s Fixed-Point Condition
Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=7, b=19. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=0. Give (modulus,a,b,x,T(x),T(x)=x?).
View answer and explanation
(7, 7, 19, 0, 5, No)
Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, 7, 19, 0, 5, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate
16. One State’s Fixed-Point Condition
Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=8, b=21. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=7. Give (modulus,a,b,x,T(x),T(x)=x?).
View answer and explanation
(8, 8, 21, 7, 5, No)
Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, 8, 21, 7, 5, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate
17. One State’s Fixed-Point Condition
Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=9, b=23. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=7. Give (modulus,a,b,x,T(x),T(x)=x?).
View answer and explanation
(9, 9, 23, 7, 5, No)
Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, 9, 23, 7, 5, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate
18. One State’s Fixed-Point Condition
Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=10, b=25. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=7. Give (modulus,a,b,x,T(x),T(x)=x?).
View answer and explanation
(10, 10, 25, 7, 5, No)
Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, 10, 25, 7, 5, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate
19. One State’s Fixed-Point Condition
Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=11, b=27. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=3. Give (modulus,a,b,x,T(x),T(x)=x?).
View answer and explanation
(5, 11, 27, 3, 0, No)
Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 11, 27, 3, 0, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate
20. One State’s Fixed-Point Condition
Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=12, b=29. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=1. Give (modulus,a,b,x,T(x),T(x)=x?).
View answer and explanation
(6, 12, 29, 1, 5, No)
Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, 12, 29, 1, 5, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate
21. One State’s Fixed-Point Condition
Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=13, b=31. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=6. Give (modulus,a,b,x,T(x),T(x)=x?).
View answer and explanation
(7, 13, 31, 6, 4, No)
Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, 13, 31, 6, 4, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate
22. One State’s Fixed-Point Condition
Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=14, b=33. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=5. Give (modulus,a,b,x,T(x),T(x)=x?).
View answer and explanation
(8, 14, 33, 5, 7, No)
Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, 14, 33, 5, 7, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate
23. One State’s Fixed-Point Condition
Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=15, b=35. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=4. Give (modulus,a,b,x,T(x),T(x)=x?).
View answer and explanation
(9, 15, 35, 4, 5, No)
Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, 15, 35, 4, 5, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate
24. One State’s Fixed-Point Condition
Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=16, b=37. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=3. Give (modulus,a,b,x,T(x),T(x)=x?).
View answer and explanation
(10, 16, 37, 3, 5, No)
Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, 16, 37, 3, 5, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate
25. One State’s Fixed-Point Condition
Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=17, b=39. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=4. Give (modulus,a,b,x,T(x),T(x)=x?).
View answer and explanation
(5, 17, 39, 4, 2, No)
Evaluate one update exactly; a fixed point has least positive period 1 in this finite self-map. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 17, 39, 4, 2, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-fixed-audit · candidate
26. A Return Time Need Not Be the Least Positive Period
Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=-3, b=-8. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=0, proposed positive return time p=1. Give (modulus,a,b,x,p,T^p(x),returns to x?,least positive period or null if x is not periodic,is the least period exactly p?).
View answer and explanation
(5, -3, -8, 0, 1, 2, No, 4, No)
Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, -3, -8, 0, 1, 2, No, 4, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate
27. A Return Time Need Not Be the Least Positive Period
Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=-2, b=-7. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=1, proposed positive return time p=2. Give (modulus,a,b,x,p,T^p(x),returns to x?,least positive period or null if x is not periodic,is the least period exactly p?).
View answer and explanation
(6, -2, -7, 1, 2, 5, No, 3, No)
Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, -2, -7, 1, 2, 5, No, 3, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate
28. A Return Time Need Not Be the Least Positive Period
Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=-1, b=-6. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=2, proposed positive return time p=3. Give (modulus,a,b,x,p,T^p(x),returns to x?,least positive period or null if x is not periodic,is the least period exactly p?).
View answer and explanation
(7, -1, -6, 2, 3, 6, No, 2, No)
Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, -1, -6, 2, 3, 6, No, 2, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate
29. A Return Time Need Not Be the Least Positive Period
Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=0, b=-5. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=3, proposed positive return time p=4. Give (modulus,a,b,x,p,T^p(x),returns to x?,least positive period or null if x is not periodic,is the least period exactly p?).
View answer and explanation
(8, 0, -5, 3, 4, 3, Yes, 1, No)
Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, 0, -5, 3, 4, 3, Yes, 1, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate
30. A Return Time Need Not Be the Least Positive Period
Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=1, b=-4. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=4, proposed positive return time p=5. Give (modulus,a,b,x,p,T^p(x),returns to x?,least positive period or null if x is not periodic,is the least period exactly p?).
View answer and explanation
(9, 1, -4, 4, 5, 2, No, 9, No)
Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, 1, -4, 4, 5, 2, No, 9, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate
31. A Return Time Need Not Be the Least Positive Period
Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=2, b=-3. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=5, proposed positive return time p=6. Give (modulus,a,b,x,p,T^p(x),returns to x?,least positive period or null if x is not periodic,is the least period exactly p?).
View answer and explanation
(10, 2, -3, 5, 6, 1, No, 4, No)
Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, 2, -3, 5, 6, 1, No, 4, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate
32. A Return Time Need Not Be the Least Positive Period
Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=3, b=-2. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=1, proposed positive return time p=7. Give (modulus,a,b,x,p,T^p(x),returns to x?,least positive period or null if x is not periodic,is the least period exactly p?).
View answer and explanation
(5, 3, -2, 1, 7, 1, Yes, 1, No)
Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 3, -2, 1, 7, 1, Yes, 1, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate
33. A Return Time Need Not Be the Least Positive Period
Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=-3, b=-1. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=1, proposed positive return time p=8. Give (modulus,a,b,x,p,T^p(x),returns to x?,least positive period or null if x is not periodic,is the least period exactly p?).
View answer and explanation
(6, -3, -1, 1, 8, 5, No, null, No)
Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, -3, -1, 1, 8, 5, No, null, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate
34. A Return Time Need Not Be the Least Positive Period
Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=-2, b=0. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=1, proposed positive return time p=1. Give (modulus,a,b,x,p,T^p(x),returns to x?,least positive period or null if x is not periodic,is the least period exactly p?).
View answer and explanation
(7, -2, 0, 1, 1, 5, No, 6, No)
Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, -2, 0, 1, 1, 5, No, 6, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate
35. A Return Time Need Not Be the Least Positive Period
Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=-1, b=1. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=1, proposed positive return time p=2. Give (modulus,a,b,x,p,T^p(x),returns to x?,least positive period or null if x is not periodic,is the least period exactly p?).
View answer and explanation
(8, -1, 1, 1, 2, 1, Yes, 2, Yes)
Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, -1, 1, 1, 2, 1, Yes, 2, Yes).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate
36. A Return Time Need Not Be the Least Positive Period
Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=0, b=2. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=1, proposed positive return time p=3. Give (modulus,a,b,x,p,T^p(x),returns to x?,least positive period or null if x is not periodic,is the least period exactly p?).
View answer and explanation
(9, 0, 2, 1, 3, 2, No, null, No)
Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, 0, 2, 1, 3, 2, No, null, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate
37. A Return Time Need Not Be the Least Positive Period
Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=1, b=3. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=1, proposed positive return time p=4. Give (modulus,a,b,x,p,T^p(x),returns to x?,least positive period or null if x is not periodic,is the least period exactly p?).
View answer and explanation
(10, 1, 3, 1, 4, 3, No, 10, No)
Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, 1, 3, 1, 4, 3, No, 10, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate
38. A Return Time Need Not Be the Least Positive Period
Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=2, b=4. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=2, proposed positive return time p=5. Give (modulus,a,b,x,p,T^p(x),returns to x?,least positive period or null if x is not periodic,is the least period exactly p?).
View answer and explanation
(5, 2, 4, 2, 5, 3, No, 4, No)
Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 2, 4, 2, 5, 3, No, 4, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate
39. A Return Time Need Not Be the Least Positive Period
Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=3, b=5. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=1, proposed positive return time p=6. Give (modulus,a,b,x,p,T^p(x),returns to x?,least positive period or null if x is not periodic,is the least period exactly p?).
View answer and explanation
(6, 3, 5, 1, 6, 5, No, null, No)
Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, 3, 5, 1, 6, 5, No, null, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate
40. A Return Time Need Not Be the Least Positive Period
Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=-3, b=6. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=0, proposed positive return time p=7. Give (modulus,a,b,x,p,T^p(x),returns to x?,least positive period or null if x is not periodic,is the least period exactly p?).
View answer and explanation
(7, -3, 6, 0, 7, 6, No, 3, No)
Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, -3, 6, 0, 7, 6, No, 3, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate
41. A Return Time Need Not Be the Least Positive Period
Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=-2, b=7. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=7, proposed positive return time p=8. Give (modulus,a,b,x,p,T^p(x),returns to x?,least positive period or null if x is not periodic,is the least period exactly p?).
View answer and explanation
(8, -2, 7, 7, 8, 5, No, null, No)
Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, -2, 7, 7, 8, 5, No, null, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate
42. A Return Time Need Not Be the Least Positive Period
Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=-1, b=8. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=7, proposed positive return time p=1. Give (modulus,a,b,x,p,T^p(x),returns to x?,least positive period or null if x is not periodic,is the least period exactly p?).
View answer and explanation
(9, -1, 8, 7, 1, 1, No, 2, No)
Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, -1, 8, 7, 1, 1, No, 2, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate
43. A Return Time Need Not Be the Least Positive Period
Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=0, b=9. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=7, proposed positive return time p=2. Give (modulus,a,b,x,p,T^p(x),returns to x?,least positive period or null if x is not periodic,is the least period exactly p?).
View answer and explanation
(10, 0, 9, 7, 2, 9, No, null, No)
Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, 0, 9, 7, 2, 9, No, null, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate
44. A Return Time Need Not Be the Least Positive Period
Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=1, b=10. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=3, proposed positive return time p=3. Give (modulus,a,b,x,p,T^p(x),returns to x?,least positive period or null if x is not periodic,is the least period exactly p?).
View answer and explanation
(5, 1, 10, 3, 3, 3, Yes, 1, No)
Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 1, 10, 3, 3, 3, Yes, 1, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate
45. A Return Time Need Not Be the Least Positive Period
Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=2, b=11. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=1, proposed positive return time p=4. Give (modulus,a,b,x,p,T^p(x),returns to x?,least positive period or null if x is not periodic,is the least period exactly p?).
View answer and explanation
(6, 2, 11, 1, 4, 1, Yes, 1, No)
Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, 2, 11, 1, 4, 1, Yes, 1, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate
46. A Return Time Need Not Be the Least Positive Period
Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=3, b=12. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=6, proposed positive return time p=5. Give (modulus,a,b,x,p,T^p(x),returns to x?,least positive period or null if x is not periodic,is the least period exactly p?).
View answer and explanation
(7, 3, 12, 6, 5, 5, No, 6, No)
Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, 3, 12, 6, 5, 5, No, 6, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate
47. A Return Time Need Not Be the Least Positive Period
Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=-3, b=13. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=5, proposed positive return time p=6. Give (modulus,a,b,x,p,T^p(x),returns to x?,least positive period or null if x is not periodic,is the least period exactly p?).
View answer and explanation
(8, -3, 13, 5, 6, 7, No, 8, No)
Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, -3, 13, 5, 6, 7, No, 8, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate
48. A Return Time Need Not Be the Least Positive Period
Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=-2, b=14. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=4, proposed positive return time p=7. Give (modulus,a,b,x,p,T^p(x),returns to x?,least positive period or null if x is not periodic,is the least period exactly p?).
View answer and explanation
(9, -2, 14, 4, 7, 0, No, 9, No)
Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, -2, 14, 4, 7, 0, No, 9, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate
49. A Return Time Need Not Be the Least Positive Period
Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=-1, b=15. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=3, proposed positive return time p=8. Give (modulus,a,b,x,p,T^p(x),returns to x?,least positive period or null if x is not periodic,is the least period exactly p?).
View answer and explanation
(10, -1, 15, 3, 8, 3, Yes, 2, No)
Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, -1, 15, 3, 8, 3, Yes, 2, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate
50. A Return Time Need Not Be the Least Positive Period
Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=0, b=16. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=4, proposed positive return time p=1. Give (modulus,a,b,x,p,T^p(x),returns to x?,least positive period or null if x is not periodic,is the least period exactly p?).
View answer and explanation
(5, 0, 16, 4, 1, 1, No, null, No)
Iterate p times, then separately enumerate to the first repeat to determine the least period. Entering a later cycle does not make the starting point periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 0, 16, 4, 1, 1, No, null, No).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-return-time-audit · candidate
51. First Repeat, Transient Length and the Starting Point’s Period
Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=-3, b=-8. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=1. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).
View answer and explanation
(5, -3, -8, 1, [1, 4, 0, 2], 1, 0, 4)
Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, -3, -8, 1, [1, 4, 0, 2], 1, 0, 4).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate
52. First Repeat, Transient Length and the Starting Point’s Period
Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=-2, b=-7. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=3. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).
View answer and explanation
(6, -2, -7, 3, [3, 5, 1], 3, 0, 3)
Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, -2, -7, 3, [3, 5, 1], 3, 0, 3).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate
53. First Repeat, Transient Length and the Starting Point’s Period
Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=-1, b=-6. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=5. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).
View answer and explanation
(7, -1, -6, 5, [5, 3], 5, 0, 2)
Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, -1, -6, 5, [5, 3], 5, 0, 2).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate
54. First Repeat, Transient Length and the Starting Point’s Period
Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=0, b=-5. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=7. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).
View answer and explanation
(8, 0, -5, 7, [7, 3], 3, 1, nonperiodic-point)
Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, 0, -5, 7, [7, 3], 3, 1, nonperiodic-point).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate
55. First Repeat, Transient Length and the Starting Point’s Period
Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=1, b=-4. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=0. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).
View answer and explanation
(9, 1, -4, 0, [0, 5, 1, 6, 2, 7, 3, 8, 4], 0, 0, 9)
Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, 1, -4, 0, [0, 5, 1, 6, 2, 7, 3, 8, 4], 0, 0, 9).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate
56. First Repeat, Transient Length and the Starting Point’s Period
Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=2, b=-3. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=1. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).
View answer and explanation
(10, 2, -3, 1, [1, 9, 5, 7], 1, 0, 4)
Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, 2, -3, 1, [1, 9, 5, 7], 1, 0, 4).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate
57. First Repeat, Transient Length and the Starting Point’s Period
Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=3, b=-2. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=3. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).
View answer and explanation
(5, 3, -2, 3, [3, 2, 4, 0], 3, 0, 4)
Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 3, -2, 3, [3, 2, 4, 0], 3, 0, 4).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate
58. First Repeat, Transient Length and the Starting Point’s Period
Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=-3, b=-1. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=3. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).
View answer and explanation
(6, -3, -1, 3, [3, 2, 5], 2, 1, nonperiodic-point)
Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, -3, -1, 3, [3, 2, 5], 2, 1, nonperiodic-point).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate
59. First Repeat, Transient Length and the Starting Point’s Period
Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=-2, b=0. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=3. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).
View answer and explanation
(7, -2, 0, 3, [3, 1, 5, 4, 6, 2], 3, 0, 6)
Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, -2, 0, 3, [3, 1, 5, 4, 6, 2], 3, 0, 6).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate
60. First Repeat, Transient Length and the Starting Point’s Period
Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=-1, b=1. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=3. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).
View answer and explanation
(8, -1, 1, 3, [3, 6], 3, 0, 2)
Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, -1, 1, 3, [3, 6], 3, 0, 2).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate
61. First Repeat, Transient Length and the Starting Point’s Period
Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=0, b=2. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=3. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).
View answer and explanation
(9, 0, 2, 3, [3, 2], 2, 1, nonperiodic-point)
Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, 0, 2, 3, [3, 2], 2, 1, nonperiodic-point).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate
62. First Repeat, Transient Length and the Starting Point’s Period
Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=1, b=3. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=3. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).
View answer and explanation
(10, 1, 3, 3, [3, 6, 9, 2, 5, 8, 1, 4, 7, 0], 3, 0, 10)
Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, 1, 3, 3, [3, 6, 9, 2, 5, 8, 1, 4, 7, 0], 3, 0, 10).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate
63. First Repeat, Transient Length and the Starting Point’s Period
Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=2, b=4. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=0. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).
View answer and explanation
(5, 2, 4, 0, [0, 4, 2, 3], 0, 0, 4)
Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 2, 4, 0, [0, 4, 2, 3], 0, 0, 4).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate
64. First Repeat, Transient Length and the Starting Point’s Period
Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=3, b=5. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=3. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).
View answer and explanation
(6, 3, 5, 3, [3, 2, 5], 2, 1, nonperiodic-point)
Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, 3, 5, 3, [3, 2, 5], 2, 1, nonperiodic-point).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate
65. First Repeat, Transient Length and the Starting Point’s Period
Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=-3, b=6. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=1. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).
View answer and explanation
(7, -3, 6, 1, [1, 3, 4], 1, 0, 3)
Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, -3, 6, 1, [1, 3, 4], 1, 0, 3).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate
66. First Repeat, Transient Length and the Starting Point’s Period
Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=-2, b=7. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=7. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).
View answer and explanation
(8, -2, 7, 7, [7, 1, 5], 5, 2, nonperiodic-point)
Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, -2, 7, 7, [7, 1, 5], 5, 2, nonperiodic-point).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate
67. First Repeat, Transient Length and the Starting Point’s Period
Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=-1, b=8. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=6. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).
View answer and explanation
(9, -1, 8, 6, [6, 2], 6, 0, 2)
Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, -1, 8, 6, [6, 2], 6, 0, 2).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate
68. First Repeat, Transient Length and the Starting Point’s Period
Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=0, b=9. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=5. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).
View answer and explanation
(10, 0, 9, 5, [5, 9], 9, 1, nonperiodic-point)
Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, 0, 9, 5, [5, 9], 9, 1, nonperiodic-point).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate
69. First Repeat, Transient Length and the Starting Point’s Period
Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=1, b=10. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=2. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).
View answer and explanation
(5, 1, 10, 2, [2], 2, 0, 1)
Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 1, 10, 2, [2], 2, 0, 1).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate
70. First Repeat, Transient Length and the Starting Point’s Period
Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=2, b=11. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=3. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).
View answer and explanation
(6, 2, 11, 3, [3, 5], 3, 0, 2)
Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, 2, 11, 3, [3, 5], 3, 0, 2).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate
71. First Repeat, Transient Length and the Starting Point’s Period
Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=3, b=12. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=6. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).
View answer and explanation
(7, 3, 12, 6, [6, 2, 4, 3, 0, 5], 6, 0, 6)
Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, 3, 12, 6, [6, 2, 4, 3, 0, 5], 6, 0, 6).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate
72. First Repeat, Transient Length and the Starting Point’s Period
Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=-3, b=13. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=3. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).
View answer and explanation
(8, -3, 13, 3, [3, 4, 1, 2, 7, 0, 5, 6], 3, 0, 8)
Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, -3, 13, 3, [3, 4, 1, 2, 7, 0, 5, 6], 3, 0, 8).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate
73. First Repeat, Transient Length and the Starting Point’s Period
Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=-2, b=14. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=0. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).
View answer and explanation
(9, -2, 14, 0, [0, 5, 4, 6, 2, 1, 3, 8, 7], 0, 0, 9)
Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, -2, 14, 0, [0, 5, 4, 6, 2, 1, 3, 8, 7], 0, 0, 9).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate
74. First Repeat, Transient Length and the Starting Point’s Period
Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=-1, b=15. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=7. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).
View answer and explanation
(10, -1, 15, 7, [7, 8], 7, 0, 2)
Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, -1, 15, 7, [7, 8], 7, 0, 2).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate
75. First Repeat, Transient Length and the Starting Point’s Period
Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=0, b=16. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=4. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).
View answer and explanation
(5, 0, 16, 4, [4, 1], 1, 1, nonperiodic-point)
Stop immediately before appending the first repeated state. Only cycle-start index 0 makes the original point periodic; otherwise it is eventually periodic but not itself periodic. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 0, 16, 4, [4, 1], 1, 1, nonperiodic-point).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-least-period · candidate
76. Every Point’s Least Period in the Complete Finite Model
Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=-3, b=-8. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. Positive period bound=1. Give (modulus,a,b,bound,every [point,least period or nonperiodic-point] row in point order,points whose least positive period is at most the bound).
View answer and explanation
(5, -3, -8, 1, [[0, 4], [1, 4], [2, 4], [3, 1], [4, 4]], [3])
Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, -3, -8, 1, [[0, 4], [1, 4], [2, 4], [3, 1], [4, 4]], [3]).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate
77. Every Point’s Least Period in the Complete Finite Model
Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=-2, b=-7. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. Positive period bound=2. Give (modulus,a,b,bound,every [point,least period or nonperiodic-point] row in point order,points whose least positive period is at most the bound).
View answer and explanation
(6, -2, -7, 2, [[0, nonperiodic-point], [1, 3], [2, nonperiodic-point], [3, 3], [4, nonperiodic-point], [5, 3]], [])
Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, -2, -7, 2, [[0, nonperiodic-point], [1, 3], [2, nonperiodic-point], [3, 3], [4, nonperiodic-point], [5, 3]], []).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate
78. Every Point’s Least Period in the Complete Finite Model
Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=-1, b=-6. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. Positive period bound=3. Give (modulus,a,b,bound,every [point,least period or nonperiodic-point] row in point order,points whose least positive period is at most the bound).
View answer and explanation
(7, -1, -6, 3, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 1], [5, 2], [6, 2]], [0, 1, 2, 3, 4, 5, 6])
Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, -1, -6, 3, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 1], [5, 2], [6, 2]], [0, 1, 2, 3, 4, 5, 6]).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate
79. Every Point’s Least Period in the Complete Finite Model
Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=0, b=-5. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. Positive period bound=4. Give (modulus,a,b,bound,every [point,least period or nonperiodic-point] row in point order,points whose least positive period is at most the bound).
View answer and explanation
(8, 0, -5, 4, [[0, nonperiodic-point], [1, nonperiodic-point], [2, nonperiodic-point], [3, 1], [4, nonperiodic-point], [5, nonperiodic-point], [6, nonperiodic-point], [7, nonperiodic-point]], [3])
Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, 0, -5, 4, [[0, nonperiodic-point], [1, nonperiodic-point], [2, nonperiodic-point], [3, 1], [4, nonperiodic-point], [5, nonperiodic-point], [6, nonperiodic-point], [7, nonperiodic-point]], [3]).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate
80. Every Point’s Least Period in the Complete Finite Model
Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=1, b=-4. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. Positive period bound=5. Give (modulus,a,b,bound,every [point,least period or nonperiodic-point] row in point order,points whose least positive period is at most the bound).
View answer and explanation
(9, 1, -4, 5, [[0, 9], [1, 9], [2, 9], [3, 9], [4, 9], [5, 9], [6, 9], [7, 9], [8, 9]], [])
Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, 1, -4, 5, [[0, 9], [1, 9], [2, 9], [3, 9], [4, 9], [5, 9], [6, 9], [7, 9], [8, 9]], []).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate
81. Every Point’s Least Period in the Complete Finite Model
Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=2, b=-3. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. Positive period bound=6. Give (modulus,a,b,bound,every [point,least period or nonperiodic-point] row in point order,points whose least positive period is at most the bound).
View answer and explanation
(10, 2, -3, 6, [[0, nonperiodic-point], [1, 4], [2, nonperiodic-point], [3, 1], [4, nonperiodic-point], [5, 4], [6, nonperiodic-point], [7, 4], [8, nonperiodic-point], [9, 4]], [1, 3, 5, 7, 9])
Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, 2, -3, 6, [[0, nonperiodic-point], [1, 4], [2, nonperiodic-point], [3, 1], [4, nonperiodic-point], [5, 4], [6, nonperiodic-point], [7, 4], [8, nonperiodic-point], [9, 4]], [1, 3, 5, 7, 9]).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate
82. Every Point’s Least Period in the Complete Finite Model
Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=3, b=-2. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. Positive period bound=1. Give (modulus,a,b,bound,every [point,least period or nonperiodic-point] row in point order,points whose least positive period is at most the bound).
View answer and explanation
(5, 3, -2, 1, [[0, 4], [1, 1], [2, 4], [3, 4], [4, 4]], [1])
Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 3, -2, 1, [[0, 4], [1, 1], [2, 4], [3, 4], [4, 4]], [1]).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate
83. Every Point’s Least Period in the Complete Finite Model
Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=-3, b=-1. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. Positive period bound=2. Give (modulus,a,b,bound,every [point,least period or nonperiodic-point] row in point order,points whose least positive period is at most the bound).
View answer and explanation
(6, -3, -1, 2, [[0, nonperiodic-point], [1, nonperiodic-point], [2, 2], [3, nonperiodic-point], [4, nonperiodic-point], [5, 2]], [2, 5])
Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, -3, -1, 2, [[0, nonperiodic-point], [1, nonperiodic-point], [2, 2], [3, nonperiodic-point], [4, nonperiodic-point], [5, 2]], [2, 5]).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate
84. Every Point’s Least Period in the Complete Finite Model
Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=-2, b=0. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. Positive period bound=3. Give (modulus,a,b,bound,every [point,least period or nonperiodic-point] row in point order,points whose least positive period is at most the bound).
View answer and explanation
(7, -2, 0, 3, [[0, 1], [1, 6], [2, 6], [3, 6], [4, 6], [5, 6], [6, 6]], [0])
Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, -2, 0, 3, [[0, 1], [1, 6], [2, 6], [3, 6], [4, 6], [5, 6], [6, 6]], [0]).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate
85. Every Point’s Least Period in the Complete Finite Model
Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=-1, b=1. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. Positive period bound=4. Give (modulus,a,b,bound,every [point,least period or nonperiodic-point] row in point order,points whose least positive period is at most the bound).
View answer and explanation
(8, -1, 1, 4, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2]], [0, 1, 2, 3, 4, 5, 6, 7])
Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, -1, 1, 4, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2]], [0, 1, 2, 3, 4, 5, 6, 7]).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate
86. Every Point’s Least Period in the Complete Finite Model
Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=0, b=2. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. Positive period bound=5. Give (modulus,a,b,bound,every [point,least period or nonperiodic-point] row in point order,points whose least positive period is at most the bound).
View answer and explanation
(9, 0, 2, 5, [[0, nonperiodic-point], [1, nonperiodic-point], [2, 1], [3, nonperiodic-point], [4, nonperiodic-point], [5, nonperiodic-point], [6, nonperiodic-point], [7, nonperiodic-point], [8, nonperiodic-point]], [2])
Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, 0, 2, 5, [[0, nonperiodic-point], [1, nonperiodic-point], [2, 1], [3, nonperiodic-point], [4, nonperiodic-point], [5, nonperiodic-point], [6, nonperiodic-point], [7, nonperiodic-point], [8, nonperiodic-point]], [2]).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate
87. Every Point’s Least Period in the Complete Finite Model
Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=1, b=3. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. Positive period bound=6. Give (modulus,a,b,bound,every [point,least period or nonperiodic-point] row in point order,points whose least positive period is at most the bound).
View answer and explanation
(10, 1, 3, 6, [[0, 10], [1, 10], [2, 10], [3, 10], [4, 10], [5, 10], [6, 10], [7, 10], [8, 10], [9, 10]], [])
Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, 1, 3, 6, [[0, 10], [1, 10], [2, 10], [3, 10], [4, 10], [5, 10], [6, 10], [7, 10], [8, 10], [9, 10]], []).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate
88. Every Point’s Least Period in the Complete Finite Model
Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=2, b=4. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. Positive period bound=1. Give (modulus,a,b,bound,every [point,least period or nonperiodic-point] row in point order,points whose least positive period is at most the bound).
View answer and explanation
(5, 2, 4, 1, [[0, 4], [1, 1], [2, 4], [3, 4], [4, 4]], [1])
Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 2, 4, 1, [[0, 4], [1, 1], [2, 4], [3, 4], [4, 4]], [1]).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate
89. Every Point’s Least Period in the Complete Finite Model
Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=3, b=5. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. Positive period bound=2. Give (modulus,a,b,bound,every [point,least period or nonperiodic-point] row in point order,points whose least positive period is at most the bound).
View answer and explanation
(6, 3, 5, 2, [[0, nonperiodic-point], [1, nonperiodic-point], [2, 2], [3, nonperiodic-point], [4, nonperiodic-point], [5, 2]], [2, 5])
Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, 3, 5, 2, [[0, nonperiodic-point], [1, nonperiodic-point], [2, 2], [3, nonperiodic-point], [4, nonperiodic-point], [5, 2]], [2, 5]).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate
90. Every Point’s Least Period in the Complete Finite Model
Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=-3, b=6. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. Positive period bound=3. Give (modulus,a,b,bound,every [point,least period or nonperiodic-point] row in point order,points whose least positive period is at most the bound).
View answer and explanation
(7, -3, 6, 3, [[0, 3], [1, 3], [2, 3], [3, 3], [4, 3], [5, 1], [6, 3]], [0, 1, 2, 3, 4, 5, 6])
Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, -3, 6, 3, [[0, 3], [1, 3], [2, 3], [3, 3], [4, 3], [5, 1], [6, 3]], [0, 1, 2, 3, 4, 5, 6]).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate
91. Every Point’s Least Period in the Complete Finite Model
Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=-2, b=7. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. Positive period bound=4. Give (modulus,a,b,bound,every [point,least period or nonperiodic-point] row in point order,points whose least positive period is at most the bound).
View answer and explanation
(8, -2, 7, 4, [[0, nonperiodic-point], [1, nonperiodic-point], [2, nonperiodic-point], [3, nonperiodic-point], [4, nonperiodic-point], [5, 1], [6, nonperiodic-point], [7, nonperiodic-point]], [5])
Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, -2, 7, 4, [[0, nonperiodic-point], [1, nonperiodic-point], [2, nonperiodic-point], [3, nonperiodic-point], [4, nonperiodic-point], [5, 1], [6, nonperiodic-point], [7, nonperiodic-point]], [5]).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate
92. Every Point’s Least Period in the Complete Finite Model
Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=-1, b=8. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. Positive period bound=5. Give (modulus,a,b,bound,every [point,least period or nonperiodic-point] row in point order,points whose least positive period is at most the bound).
View answer and explanation
(9, -1, 8, 5, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 1], [5, 2], [6, 2], [7, 2], [8, 2]], [0, 1, 2, 3, 4, 5, 6, 7, 8])
Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, -1, 8, 5, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 1], [5, 2], [6, 2], [7, 2], [8, 2]], [0, 1, 2, 3, 4, 5, 6, 7, 8]).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate
93. Every Point’s Least Period in the Complete Finite Model
Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=0, b=9. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. Positive period bound=6. Give (modulus,a,b,bound,every [point,least period or nonperiodic-point] row in point order,points whose least positive period is at most the bound).
View answer and explanation
(10, 0, 9, 6, [[0, nonperiodic-point], [1, nonperiodic-point], [2, nonperiodic-point], [3, nonperiodic-point], [4, nonperiodic-point], [5, nonperiodic-point], [6, nonperiodic-point], [7, nonperiodic-point], [8, nonperiodic-point], [9, 1]], [9])
Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, 0, 9, 6, [[0, nonperiodic-point], [1, nonperiodic-point], [2, nonperiodic-point], [3, nonperiodic-point], [4, nonperiodic-point], [5, nonperiodic-point], [6, nonperiodic-point], [7, nonperiodic-point], [8, nonperiodic-point], [9, 1]], [9]).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate
94. Every Point’s Least Period in the Complete Finite Model
Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=1, b=10. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. Positive period bound=1. Give (modulus,a,b,bound,every [point,least period or nonperiodic-point] row in point order,points whose least positive period is at most the bound).
View answer and explanation
(5, 1, 10, 1, [[0, 1], [1, 1], [2, 1], [3, 1], [4, 1]], [0, 1, 2, 3, 4])
Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 1, 10, 1, [[0, 1], [1, 1], [2, 1], [3, 1], [4, 1]], [0, 1, 2, 3, 4]).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate
95. Every Point’s Least Period in the Complete Finite Model
Use the complete finite self-map on Fin(6)={0,…,5}, T(x)=(a*x+b) mod 6, a=2, b=11. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. Positive period bound=2. Give (modulus,a,b,bound,every [point,least period or nonperiodic-point] row in point order,points whose least positive period is at most the bound).
View answer and explanation
(6, 2, 11, 2, [[0, nonperiodic-point], [1, 1], [2, nonperiodic-point], [3, 2], [4, nonperiodic-point], [5, 2]], [1, 3, 5])
Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, 2, 11, 2, [[0, nonperiodic-point], [1, 1], [2, nonperiodic-point], [3, 2], [4, nonperiodic-point], [5, 2]], [1, 3, 5]).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate
96. Every Point’s Least Period in the Complete Finite Model
Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=3, b=12. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. Positive period bound=3. Give (modulus,a,b,bound,every [point,least period or nonperiodic-point] row in point order,points whose least positive period is at most the bound).
View answer and explanation
(7, 3, 12, 3, [[0, 6], [1, 1], [2, 6], [3, 6], [4, 6], [5, 6], [6, 6]], [1])
Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, 3, 12, 3, [[0, 6], [1, 1], [2, 6], [3, 6], [4, 6], [5, 6], [6, 6]], [1]).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate
97. Every Point’s Least Period in the Complete Finite Model
Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=-3, b=13. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. Positive period bound=4. Give (modulus,a,b,bound,every [point,least period or nonperiodic-point] row in point order,points whose least positive period is at most the bound).
View answer and explanation
(8, -3, 13, 4, [[0, 8], [1, 8], [2, 8], [3, 8], [4, 8], [5, 8], [6, 8], [7, 8]], [])
Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, -3, 13, 4, [[0, 8], [1, 8], [2, 8], [3, 8], [4, 8], [5, 8], [6, 8], [7, 8]], []).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate
98. Every Point’s Least Period in the Complete Finite Model
Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=-2, b=14. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. Positive period bound=5. Give (modulus,a,b,bound,every [point,least period or nonperiodic-point] row in point order,points whose least positive period is at most the bound).
View answer and explanation
(9, -2, 14, 5, [[0, 9], [1, 9], [2, 9], [3, 9], [4, 9], [5, 9], [6, 9], [7, 9], [8, 9]], [])
Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, -2, 14, 5, [[0, 9], [1, 9], [2, 9], [3, 9], [4, 9], [5, 9], [6, 9], [7, 9], [8, 9]], []).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate
99. Every Point’s Least Period in the Complete Finite Model
Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=-1, b=15. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. Positive period bound=6. Give (modulus,a,b,bound,every [point,least period or nonperiodic-point] row in point order,points whose least positive period is at most the bound).
View answer and explanation
(10, -1, 15, 6, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2]], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, -1, 15, 6, [[0, 2], [1, 2], [2, 2], [3, 2], [4, 2], [5, 2], [6, 2], [7, 2], [8, 2], [9, 2]], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate
100. Every Point’s Least Period in the Complete Finite Model
Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=0, b=16. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. Positive period bound=1. Give (modulus,a,b,bound,every [point,least period or nonperiodic-point] row in point order,points whose least positive period is at most the bound).
View answer and explanation
(5, 0, 16, 1, [[0, nonperiodic-point], [1, 1], [2, nonperiodic-point], [3, nonperiodic-point], [4, nonperiodic-point]], [1])
Enumerate every state, distinguish transient starting points from cycle points and filter only defined least periods. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 0, 16, 1, [[0, nonperiodic-point], [1, 1], [2, nonperiodic-point], [3, nonperiodic-point], [4, nonperiodic-point]], [1]).
Reviewed step-by-step derivation
Orbits, fixed points and periodic points · qf-app21-orbit-periodic-inventory · candidate
Canonical question JSONL · English JSONL · Validation report