Question 1566: A Fixed-Length Discrete Orbit Prefix

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=16, b=38. 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. x0=0, updates=6. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].

• G001 — Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=16, b=38. 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. x0=0, updates=6. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].

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)=(16x+(38)) mod 5

remainder representative always lies in X

x₀=0; transitions=6

initial orbit prefix=[0]

transition1: a·x := mul(16, 0)

transition1: a·x = 0

transition1: a·x+b := sum(0, 38)

transition1: a·x+b = 38

reduce 38 modulo 5

38=7·5+3

transition2: a·x := mul(16, 3)

transition2: a·x = 48

transition2: a·x+b := sum(48, 38)

transition2: a·x+b = 86

reduce 86 modulo 5

86=17·5+1

transition3: a·x := mul(16, 1)

transition3: a·x = 16

transition3: a·x+b := sum(16, 38)

transition3: a·x+b = 54

reduce 54 modulo 5

54=10·5+4

transition4: a·x := mul(16, 4)

transition4: a·x = 64

transition4: a·x+b := sum(64, 38)

transition4: a·x+b = 102

reduce 102 modulo 5

102=20·5+2

transition5: a·x := mul(16, 2)

transition5: a·x = 32

transition5: a·x+b := sum(32, 38)

transition5: a·x+b = 70

reduce 70 modulo 5

70=14·5+0

transition6: a·x := mul(16, 0)

transition6: a·x = 0

transition6: a·x+b := sum(0, 38)

transition6: a·x+b = 38

reduce 38 modulo 5

38=7·5+3

computed transitions=6

orbit=[0, 3, 1, 4, 2, 0, 3]; number of entries=7

a finite requested prefix

no unrequested least-period claim

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 transition6: a·x+b with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute transition6: 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 — Recompute transition5: a·x+b with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute transition5: a·x with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Recompute transition4: a·x+b with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute transition4: a·x with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Recompute transition3: a·x+b with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute transition3: a·x with exact rational arithmetic, not rounded decimals.
• 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 transition2: a·x+b with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute transition2: 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 — Recompute transition1: a·x+b with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute transition1: a·x 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, 16, 38, 0, 6, [0, 3, 1, 4, 2, 0, 3])

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.