Question 1525: Every Fixed Point of the Given Finite Self-Map

Original question and givens

Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=20, b=49. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).

• G001 — Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=20, b=49. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).

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-discrete-dynamical-system

Step-by-step derivation

X={0 through 4}; T(x)=(20x+(49)) mod 5

remainder representative always lies in X

candidate0: a·x := mul(20, 0)

candidate0: a·x = 0

candidate0: a·x+b := sum(0, 49)

candidate0: a·x+b = 49

reduce 49 modulo 5

49=9·5+4

T(0)=4

fixed-point condition T(x)=x: No

candidate1: a·x := mul(20, 1)

candidate1: a·x = 20

candidate1: a·x+b := sum(20, 49)

candidate1: a·x+b = 69

reduce 69 modulo 5

69=13·5+4

T(1)=4

fixed-point condition T(x)=x: No

candidate2: a·x := mul(20, 2)

candidate2: a·x = 40

candidate2: a·x+b := sum(40, 49)

candidate2: a·x+b = 89

reduce 89 modulo 5

89=17·5+4

T(2)=4

fixed-point condition T(x)=x: No

candidate3: a·x := mul(20, 3)

candidate3: a·x = 60

candidate3: a·x+b := sum(60, 49)

candidate3: a·x+b = 109

reduce 109 modulo 5

109=21·5+4

T(3)=4

fixed-point condition T(x)=x: No

candidate4: a·x := mul(20, 4)

candidate4: a·x = 80

candidate4: a·x+b := sum(80, 49)

candidate4: a·x+b = 129

reduce 129 modulo 5

129=25·5+4

T(4)=4

fixed-point condition T(x)=x: Yes

all 5 carrier elements audited

complete fixed set=[4]

Reverse checks

• S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S019 · premise_and_result_audit — Recompute candidate4: a·x+b with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute candidate4: a·x with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S015 · premise_and_result_audit — Recompute candidate3: a·x+b with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute candidate3: a·x with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 candidate2: a·x+b with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute candidate2: a·x 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Recompute candidate1: a·x+b with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute candidate1: a·x with exact rational arithmetic, not rounded decimals.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Recompute candidate0: a·x+b with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute candidate0: a·x with exact rational arithmetic, not rounded decimals.
• 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, 20, 49, [4])

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.