Question 1563: A Fixed-Length Discrete Orbit Prefix

Original question and givens

Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=13, b=29. 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=3. 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(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=13, b=29. 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=3. 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 6}; T(x)=(13x+(29)) mod 7

remainder representative always lies in X

x₀=5; transitions=3

initial orbit prefix=[5]

transition1: a·x := mul(13, 5)

transition1: a·x = 65

transition1: a·x+b := sum(65, 29)

transition1: a·x+b = 94

reduce 94 modulo 7

94=13·7+3

transition2: a·x := mul(13, 3)

transition2: a·x = 39

transition2: a·x+b := sum(39, 29)

transition2: a·x+b = 68

reduce 68 modulo 7

68=9·7+5

transition3: a·x := mul(13, 5)

transition3: a·x = 65

transition3: a·x+b := sum(65, 29)

transition3: a·x+b = 94

reduce 94 modulo 7

94=13·7+3

computed transitions=3

orbit=[5, 3, 5, 3]; number of entries=4

a finite requested prefix

no unrequested least-period claim

Reverse checks

• 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 — 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

(7, 13, 29, 5, 3, [5, 3, 5, 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.