Question 2582: A Return Time Need Not Be the Least Positive Period

Original question and givens

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?).

• G001 — 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?).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-orbit-fixed-periodic

Step-by-step derivation

n=5; a=3; b=-2

S={0,...,4}; T(z)=(a·z+b) mod n

start=1

record each state before its first successor calculation

least period step1 product := mul(3, 1)

least period step1 product = 3

least period step1 affine value := sum(3, -2)

least period step1 affine value = 1

Euclidean division of 1 by 5

1=0·5+1

distinct trace=[1]; first repeated state=1

cycle_start=0; starting-point least period=1

return audit1 product := mul(3, 1)

return audit1 product = 3

return audit1 affine value := sum(3, -2)

return audit1 affine value = 1

Euclidean division of 1 by 5

1=0·5+1

return audit2 product := mul(3, 1)

return audit2 product = 3

return audit2 affine value := sum(3, -2)

return audit2 affine value = 1

Euclidean division of 1 by 5

1=0·5+1

return audit3 product := mul(3, 1)

return audit3 product = 3

return audit3 affine value := sum(3, -2)

return audit3 affine value = 1

Euclidean division of 1 by 5

1=0·5+1

return audit4 product := mul(3, 1)

return audit4 product = 3

return audit4 affine value := sum(3, -2)

return audit4 affine value = 1

Euclidean division of 1 by 5

1=0·5+1

return audit5 product := mul(3, 1)

return audit5 product = 3

return audit5 affine value := sum(3, -2)

return audit5 affine value = 1

Euclidean division of 1 by 5

1=0·5+1

return audit6 product := mul(3, 1)

return audit6 product = 3

return audit6 affine value := sum(3, -2)

return audit6 affine value = 1

Euclidean division of 1 by 5

1=0·5+1

return audit7 product := mul(3, 1)

return audit7 product = 3

return audit7 affine value := sum(3, -2)

return audit7 affine value = 1

Euclidean division of 1 by 5

1=0·5+1

T^7(1)=1; least period=1

returns=Yes; period exactly p=No

Reverse checks

• S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S026 · premise_and_result_audit — Recompute return audit7 affine value with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute return audit7 product with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S023 · premise_and_result_audit — Recompute return audit6 affine value with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute return audit6 product with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S020 · premise_and_result_audit — Recompute return audit5 affine value with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute return audit5 product with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S017 · premise_and_result_audit — Recompute return audit4 affine value with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute return audit4 product with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Recompute return audit3 affine value with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute return audit3 product with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Recompute return audit2 affine value with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute return audit2 product with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Recompute return audit1 affine value with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute return audit1 product with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Replay the cycle of this fixed self-map;return or nonreturn is certified only for this complete finite model,not an unspecified system.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Recompute least period step1 affine value with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute least period step1 product with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(5, 3, -2, 1, 7, 1, Yes, 1, No)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.