Question 1519: 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=14, b=37. 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=14, b=37. 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)=(14x+(37)) mod 5

remainder representative always lies in X

candidate0: a·x := mul(14, 0)

candidate0: a·x = 0

candidate0: a·x+b := sum(0, 37)

candidate0: a·x+b = 37

reduce 37 modulo 5

37=7·5+2

T(0)=2

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

candidate1: a·x := mul(14, 1)

candidate1: a·x = 14

candidate1: a·x+b := sum(14, 37)

candidate1: a·x+b = 51

reduce 51 modulo 5

51=10·5+1

T(1)=1

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

candidate2: a·x := mul(14, 2)

candidate2: a·x = 28

candidate2: a·x+b := sum(28, 37)

candidate2: a·x+b = 65

reduce 65 modulo 5

65=13·5+0

T(2)=0

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

candidate3: a·x := mul(14, 3)

candidate3: a·x = 42

candidate3: a·x+b := sum(42, 37)

candidate3: a·x+b = 79

reduce 79 modulo 5

79=15·5+4

T(3)=4

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

candidate4: a·x := mul(14, 4)

candidate4: a·x = 56

candidate4: a·x+b := sum(56, 37)

candidate4: a·x+b = 93

reduce 93 modulo 5

93=18·5+3

T(4)=3

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

all 5 carrier elements audited

complete fixed set=[1]

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, 14, 37, [1])

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.