Question 1507: 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=2, b=13. 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=2, b=13. 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)=(2x+(13)) mod 5

remainder representative always lies in X

candidate0: a·x := mul(2, 0)

candidate0: a·x = 0

candidate0: a·x+b := sum(0, 13)

candidate0: a·x+b = 13

reduce 13 modulo 5

13=2·5+3

T(0)=3

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

candidate1: a·x := mul(2, 1)

candidate1: a·x = 2

candidate1: a·x+b := sum(2, 13)

candidate1: a·x+b = 15

reduce 15 modulo 5

15=3·5+0

T(1)=0

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

candidate2: a·x := mul(2, 2)

candidate2: a·x = 4

candidate2: a·x+b := sum(4, 13)

candidate2: a·x+b = 17

reduce 17 modulo 5

17=3·5+2

T(2)=2

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

candidate3: a·x := mul(2, 3)

candidate3: a·x = 6

candidate3: a·x+b := sum(6, 13)

candidate3: a·x+b = 19

reduce 19 modulo 5

19=3·5+4

T(3)=4

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

candidate4: a·x := mul(2, 4)

candidate4: a·x = 8

candidate4: a·x+b := sum(8, 13)

candidate4: a·x+b = 21

reduce 21 modulo 5

21=4·5+1

T(4)=1

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

all 5 carrier elements audited

complete fixed set=[2]

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, 2, 13, [2])

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.