Question 2579: A Return Time Need Not Be the Least Positive Period

Original question and givens

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?).

• G001 — 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?).

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=8; a=0; b=-5

S={0,...,7}; T(z)=(a·z+b) mod n

start=3

record each state before its first successor calculation

least period step1 product := mul(0, 3)

least period step1 product = 0

least period step1 affine value := sum(0, -5)

least period step1 affine value = -5

Euclidean division of -5 by 8

-5=-1·8+3

distinct trace=[3]; first repeated state=3

cycle_start=0; starting-point least period=1

return audit1 product := mul(0, 3)

return audit1 product = 0

return audit1 affine value := sum(0, -5)

return audit1 affine value = -5

Euclidean division of -5 by 8

-5=-1·8+3

return audit2 product := mul(0, 3)

return audit2 product = 0

return audit2 affine value := sum(0, -5)

return audit2 affine value = -5

Euclidean division of -5 by 8

-5=-1·8+3

return audit3 product := mul(0, 3)

return audit3 product = 0

return audit3 affine value := sum(0, -5)

return audit3 affine value = -5

Euclidean division of -5 by 8

-5=-1·8+3

return audit4 product := mul(0, 3)

return audit4 product = 0

return audit4 affine value := sum(0, -5)

return audit4 affine value = -5

Euclidean division of -5 by 8

-5=-1·8+3

T^4(3)=3; least period=1

returns=Yes; period exactly p=No

Reverse checks

• S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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

(8, 0, -5, 3, 4, 3, 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.