Question 1564: A Fixed-Length Discrete Orbit Prefix

Original question and givens

Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=14, b=32. 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=5, updates=4. 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(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=14, b=32. 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=5, updates=4. 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 7}; T(x)=(14x+(32)) mod 8

remainder representative always lies in X

x₀=5; transitions=4

initial orbit prefix=[5]

transition1: a·x := mul(14, 5)

transition1: a·x = 70

transition1: a·x+b := sum(70, 32)

transition1: a·x+b = 102

reduce 102 modulo 8

102=12·8+6

transition2: a·x := mul(14, 6)

transition2: a·x = 84

transition2: a·x+b := sum(84, 32)

transition2: a·x+b = 116

reduce 116 modulo 8

116=14·8+4

transition3: a·x := mul(14, 4)

transition3: a·x = 56

transition3: a·x+b := sum(56, 32)

transition3: a·x+b = 88

reduce 88 modulo 8

88=11·8+0

transition4: a·x := mul(14, 0)

transition4: a·x = 0

transition4: a·x+b := sum(0, 32)

transition4: a·x+b = 32

reduce 32 modulo 8

32=4·8+0

computed transitions=4

orbit=[5, 6, 4, 0, 0]; number of entries=5

a finite requested prefix

no unrequested least-period claim

Reverse checks

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

(8, 14, 32, 5, 4, [5, 6, 4, 0, 0])

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.