REVIEW 3 major objections
Finding Prime Numbers as Fixed Points of Sequences
T0 review · 3 major / 0 minor · reviewed 2026-05-24 · grok-4.3
Pith's one-line read Odd primes appear as fixed points of the sequence of smallest unused divisors of triangular numbers.
desk verdict Empirical observation that a new sequence from unused divisors of triangular numbers has fixed points matching odd primes, but no proof or bound on exceptions. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
Sequence A(1) of smallest unused divisors of triangular numbers, whose fixed points are asserted to be the odd primes.
What would settle it
A single triangular number whose smallest unused divisor is a composite that enters the sequence as a fixed point, or an odd prime that is never selected.
Extended reading notes
Core claim
The paper claims that the sequence A(1), formed by selecting at each step the smallest positive divisor of the next triangular number that has not already appeared, yields fixed points that coincide with the set of odd primes.
Load-bearing premise
The rule for selecting the next unused divisor of triangular numbers produces fixed points that match the odd primes exactly, with no systematic exceptions.
Editorial extensions
If this is right
- All odd primes arise exactly once as fixed points under this divisor rule.
- No composite numbers appear as fixed points in the sequence.
- The construction supplies an explicit, non-probabilistic procedure that recovers the odd primes from the sequence of triangular numbers.
- The method extends in principle to arbitrarily large triangular numbers while preserving the observed accuracy.
Reading between the lines
- The pattern may reflect an underlying arithmetic property of triangular numbers that forces composites to be skipped once smaller factors have been used.
- One could check whether the same fixed-point behavior appears when triangular numbers are replaced by other polygonal numbers.
- If the pattern continues without exception, the sequence could serve as a deterministic primality filter for numbers up to a given bound.
- The construction invites comparison with other divisor-based sequences that isolate primes.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript claims that odd primes arise as fixed points of the sequence A(1), constructed by taking the smallest unused positive divisor of each successive triangular number T_n = n(n+1)/2. It reports that basic calculations yield success rates above 99.9% for identifying primes this way but supplies neither a formal definition of 'fixed point' nor a proof that the construction yields precisely the odd primes.
Significance. A rigorous proof of the claimed characterization would be of interest in number theory as an alternative description of primes via divisors of triangular numbers. The present work, however, offers only an empirical observation without theoretical justification, bounds on exceptions, or analysis of the construction's completeness, so its significance remains that of an unproven conjecture.
major comments (3)
- Abstract: the phrase 'fixed points of the sequence' is never defined. It is therefore impossible to determine whether the reported 99.9% success rate refers to a well-posed mathematical property or merely an informal counting procedure.
- Abstract: the central assertion that 'the set of odd primes can be obtained as fixed points' is stated without proof, without a precise statement (e.g., surjectivity onto odd primes, injectivity, exclusion of all composites), and without any error analysis or range of n examined.
- The construction is deterministic and the triangular numbers are unbounded; the manuscript therefore requires at least a heuristic argument or explicit search bound showing that no composite can ever become the smallest unused divisor of some later T_n, yet none is supplied.
Simulated Author's Rebuttal
We thank the referee for the careful reading of our manuscript. The work is an observational note reporting computational findings on a sequence of smallest unused divisors of triangular numbers and their apparent relation to odd primes. We address each major comment below.
read point-by-point responses
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Referee: Abstract: the phrase 'fixed points of the sequence' is never defined. It is therefore impossible to determine whether the reported 99.9% success rate refers to a well-posed mathematical property or merely an informal counting procedure.
Authors: We agree that a formal definition is needed. In the revised version we will add an explicit definition of the sequence A(1) and of the term 'fixed point' as used in our calculations: a positive integer m is recorded as a fixed point when it is selected as the smallest unused divisor of some T_n and does not appear earlier in the sequence. revision: yes
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Referee: Abstract: the central assertion that 'the set of odd primes can be obtained as fixed points' is stated without proof, without a precise statement (e.g., surjectivity onto odd primes, injectivity, exclusion of all composites), and without any error analysis or range of n examined.
Authors: The manuscript uses the phrasing 'it seems that' precisely to indicate an empirical observation rather than a theorem. We will revise the abstract and introduction to state explicitly that the claim is conjectural, to report the range of n examined in our calculations, and to include any observed exceptions or error counts. revision: partial
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Referee: The construction is deterministic and the triangular numbers are unbounded; the manuscript therefore requires at least a heuristic argument or explicit search bound showing that no composite can ever become the smallest unused divisor of some later T_n, yet none is supplied.
Authors: The present note contains only computational verification up to a finite bound and does not supply a heuristic or proof that composites are excluded for all larger n. We do not have such an argument at this time. revision: no
- A rigorous proof that the construction yields exactly the odd primes (with no composites and with all odd primes appearing).
Circularity Check
No circularity; empirical observation from independent sequence definition
full rationale
The paper defines sequence A(1) solely in terms of successive smallest unused positive divisors of triangular numbers T_n = n(n+1)/2, with no reference to primes or primality in the construction itself. It then reports an empirical observation (via basic calculations) that the fixed points of this sequence coincide with odd primes at >99.9% success rate. No derivation, equation, or self-citation is supplied that reduces the claimed correspondence to the definition by construction, nor is any parameter fitted to prime data and then relabeled as a prediction. The result is therefore self-contained as an observation against an external benchmark (the set of odd primes) and receives score 0.
Assumptions & free parameters
Cite this review
Pith. "Pith review of Finding Prime Numbers as Fixed Points of Sequences." pith.science (2026). https://pith.science/paper/GKHNOPVC
@misc{pith2026190710023,
author = {Pith},
title = {Pith review of: Finding Prime Numbers as Fixed Points of Sequences},
year = {2026},
howpublished = {\url{https://pith.science/paper/GKHNOPVC}},
note = {Machine review of arXiv:1907.10023}
}
read the original abstract
In this note we describe a method for finding prime numbers as fixed points of particularly simple sequences. Some basic calculations show that success rates for identifying primes this way are over 99.9%. In particular, it seems that the set of odd primes can be obtained as fixed points of the sequence which we call A(1), the sequence of smallest divisors of triangular numbers, where the divisors are positive numbers that have not yet appeared in the sequence.
Reviewed May 24, 2026 · model on record in the stance chip above.
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