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Math-636 · mathematics · Number theory

Shanks Chains of Length 7

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

A chain of length seven is a list of seven primes $p_1,\ldots,p_7$ with $p_{i+1}=4p_i^2-17$ for $1\leq i\leq6$. The first prime is not prescribed. All seven terms must be positive primes; changing the recurrence or counting seven transitions would define a different search problem. [S1] [S2]

Statement recorded in the supplied source.
Are there any Shanks chains of length 7 with $p_{i+1} = 4p_i^2 - 17$? [S1]

Short English statement

Can seven consecutive terms of the recurrence four times the square minus seventeen all be prime?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-636-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on Math-636

A Shanks Prime Chain of Length Seven

Definitions and mathematical statement

Shared notation.
Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

A chain of length seven is a list of seven primes $p_1,\ldots,p_7$ with $p_{i+1}=4p_i^2-17$ for $1\leq i\leq6$. The first prime is not prescribed. All seven terms must be positive primes; changing the recurrence or counting seven transitions would define a different search problem. [S1] [S2]

Statement recorded in the supplied source.
Are there any Shanks chains of length 7 with $p_{i+1} = 4p_i^2 - 17$? [S1]

Short English statement

Can seven consecutive terms of the recurrence four times the square minus seventeen all be prime?

Research attempt

A catalogue-status discrepancy must be resolved first.
The supplied statement labels existence of a seven-term chain as open, but its own reference [S2] reports a length-seven example found by Paul Leyland on 22 November 2000, starting at $176996212597$, with a $758$-digit seventh term. This is a reported prior resolution of the literal existence question, not a new conjectural candidate invented here. The cited primary announcement [E1] was sought on 1 October 2026 but could not be retrieved. Thus this notebook performs a mathematical and computational audit of the reported example and records the exact limit of independent certification, rather than falsely declaring the literature question still open or claiming a new solution.
Exact reconstruction of the seven reported terms.
Let $F(t)=4t^2-17$ and set \[ p_1=176996212597,\qquad p_j=F^{j-1}(p_1)\quad(2\le j\le7). \] This specifies seven integers with six transitions, precisely matching the catalogue convention. The script executes all six integer recurrences and writes every term in full to the result file. Their decimal digit counts are \[ 12,24,47,95,189,379,758. \] The last count agrees with the catalogue reference's report. Exact agreement of recurrence and size is useful evidence of identifying the reported chain, but is not a primality certificate for the terms.
Growth and positivity are elementary.
For every integer $t\ge3$, \[ F(t)-t=4t^2-t-17>0, \] so a chain starting above two remains positive and strictly increases. Starting at the prime two fails immediately, because $F(2)=-1$. For $t\ge3$, the bounds $2t^2<F(t)<4t^2$ hold, giving \[ 2\log t+\log2<\log F(t)<2\log t+\log4. \] Thus bit length roughly doubles at each transition. This explains why checking the first few primes is qualitatively different from proving the final hundreds-of-digits term prime, despite the small number of transitions.
An exact congruence sieve for potential starts.
For a fixed prime $\ell$, reduce the map $F$ modulo $\ell$. A start residue can be rejected if one of its first seven iterates is zero modulo $\ell$, provided the corresponding integer term is larger than $\ell$. Equality to $\ell$ is a prime exception and must not be discarded by an indiscriminate divisibility test.

For example modulo thirteen, $F(t)\equiv4(t^2-1)$. The residues $\pm1$ reach zero in one step, and $\pm2$ reach $-1$ and then zero. Direct enumeration shows that the residues surviving seven nonzero terms are \[ 3,4,5,6,7,8,9,10\pmod{13}. \] For starts larger than thirteen, every actual term is larger than thirteen by growth, so this is a sound necessary condition for a seven-prime chain. For smaller starts the equality exception must be tested using the actual integer term.

For any finite list of primes one can intersect the allowed residue conditions using the Chinese remainder theorem. This is a finite search filter, not a proof of simultaneous primality. A residue surviving the small-prime filters can still yield a composite term whose factors exceed all sieve primes.

What was and was not certified locally.
The script proves $p_1$ prime by trial division through its integer square root. It reconstructs all later terms exactly, and separately enumerates seven-step nonzero residue classes for small moduli. It does not claim a deterministic primality certificate for $p_2$ through $p_7$. No probable-prime test is silently promoted to a primality proof; indeed none is needed for the stated local results. The result file explicitly marks those six primalities as not certified here.

A complete independent verification could use a checked ECPP certificate, or another rigorous primality certificate, for each later term. Merely testing many Fermat bases would not close that gap. Likewise a chain of probable primes would meet a different, weaker search specification than the seven actual primes required here.

Artifacts and literature boundary.
The exact terms and digit counts are stored in results_second.json, generated by checks_second.py, under _Local/Erdos_problems/UnsolvedMath/attempt_audit_2026_10_01/number_theory/. The source report supplies a named prior example, while the independent arithmetic verifies recurrence, positivity, term counts, and the first primality. These are distinct evidence levels and are kept separate. No claim to have contacted the original author or obtained unavailable certificates is made.
Outcome.
Prior existence reported; local certification UNRESOLVED. The literal catalogue question is reported answered by the existing source. This attempt does not independently certify all seven primes and therefore retains an unresolved attempt marker, referring to that audit gap rather than asserting a fresh open status for the published existence question. A reviewer should update catalogue status only after evaluating the original announcement or an equivalent certificate.

Sources

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