REVIEW 3 major objections 5 minor 5 references
Generalized Schatunowsky theorem in a weak arithmetic
T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read The generalized Schatunowsky theorem—that for each prime $p$ only finitely many $p$-good numbers exist—is provable from the weak base theory $\mathrm{PA}^-$ together with a few explicit axioms about primes, and one axiom is shown…
desk verdict A fixable one-character error in A17 makes the stated theories inconsistent, but the intended theorem and proof strategy are sound. 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
The argument is carried by the weak base theory $\mathrm{PA}^-$, whose models are the positive cones of discretely ordered rings, together with the choice of 'prime' as $\pi_2(x)$ (the divisibility property, not mere irreducibility). The additional axioms supply the needed number-theoretic content: A16 and A17 give successor and predecessor primes, A18 gives a largest prime below $\sqrt{n}$, and A19, the inequality $S(q)^2 < 2qP(q)$ for consecutive primes with $q\ge 19$, is the engine that forces any sufficiently large $m$ to have a composite totative. The model-theoretic proof takes a nonstandard prime $q$ with $q^2<m$, observes that $q^2$, $P(q)^2$, and $P(P(q))^2$ are totatives of $m$ unless $q$, $P(q)$, and $P(P(q))$ divide $m$, and derives a contradiction from A19. Independence results use polynomial rings $\mathbb{Z}[X]$ and $\mathbb{Q}\mathbb{Z}[X]$ as models.
What would settle it
In a nonstandard model of the corrected axioms, check whether any $p$-good number exceeds the bound $S(S(S(p)))^2+1$; the paper predicts none. One can also verify in the standard model whether any $m\ge 290$ is $p$-good for $p\in\{2,3,5\}$, which would falsify the small-prime case.
Extended reading notes
Core claim
The central discovery is that the cited generalization of Schatunowsky's theorem—that for each prime $p$ the set of $p$-good numbers is finite—is provable in the weak base theory $\mathrm{PA}^-$ augmented by a small, explicit list of axioms about primes. The key bound is that in any model of the weak theory, for a nonstandard prime $p\ge 7$, no number $m\ge S(S(S(p)))^2+1$ can be $p$-good; the argument uses the axiom A19, which states that the square of a prime $\ge 19$ is less than twice the product of the two preceding primes. For the strong form, two additional axioms (every number $>1$ has a prime divisor, and a bound on how far a prime can be from the square of the next prime) produce a largest $p$-good number. The paper also establishes an independence result: the axiom A18, asserting existence of a largest prime whose square is below $n$, is not needed for the strong form, since the positive cone of $\mathbb{Z}[X]$ satisfies the remaining axioms and the strong form.
Load-bearing premise
The load-bearing premise is the intended reading of A17 as 'every prime greater than 2 has a predecessor prime'; as written, A17 applies to all primes, including 2, which has no predecessor in the standard model, making the theory $\Sigma$ inconsistent and the model-theoretic proof vacuous.
Editorial extensions
If this is right
- In every model of $\Sigma$, for each prime $p\ge 7$ every $p$-good number is $< S(S(S(p)))^2+1$, and for $p\in\{2,3,5\}$ every $p$-good number is $< 290$.
- The strong form holds in every model of $\Sigma'$, so in every such model each prime $p$ has a largest $p$-good number.
- The axiom A18 is not needed for the strong form: the theory $\mathrm{PA}^-$ together with A16, A17, A19, A20, and A21 proves $\mathrm{GS_s}$, and there is a model satisfying these axioms in which A18 fails.
- The two finitely axiomatized systems $\Sigma$ and $\Sigma'$ show that the generalized Schatunowsky theorem is a theorem of finite elementary number theory, not a consequence of the full induction schema.
- The independence results clarify the axioms' roles: A16 and A17 are not consequences of $\mathrm{PA}^-$ with A19 and A20, and A18 is not a consequence of the other axioms together with A21.
Reading between the lines
- A natural correction to the displayed A17—requiring $2<p$—would make the axiom satisfiable, and since the proof only uses A17 for primes $\ge 17$, the corrected version should still yield the weak form.
- The proof strategy, bounding $p$-good numbers by iterating the successor-prime function and applying an inequality like A19, could serve as a template for proving other 'largest number with a given totative property' theorems inside weak arithmetic.
- Because the paper chooses $\pi_2$ (divisibility) rather than $\pi_1$ (irreducibility) for 'prime', the theorem's status in weak arithmetic depends on this choice; in models where the two notions differ, the $p$-good property might behave differently.
- The independence of A18 from the strong form suggests that an elementary proof of $\mathrm{GS_s}$ avoiding the model-theoretic argument may exist, though the paper does not supply one.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves, in a weak arithmetic base PA^- (the ordered semiring axioms A1-A15 without induction), that a generalized Schatunowsky theorem holds: Theorem 4.1 derives a weak form GSw from the finite axiom set Sigma = PA^- + A16-A19, and Theorem 4.2 derives a strong form GSs from Sigma' = PA^- + A16-A21. The added axioms are explicit elementary number-theoretic statements about successor/predecessor primes, existence of largest primes below a square, a three-prime inequality, existence of prime divisors, and a divisibility inequality. The proof for nonstandard models uses A18 to produce large primes and A19 for the key inequality; the standard model case is handled by the Kaneko-Nakai classification. Section 5 gives independence results using polynomial ring models, and Section 6 shows A18 is not needed for GSs in the sense that C(Z[X]) satisfies all axioms except A18 while GSs holds there.
Significance. If correct, the paper is a valuable contribution to reverse mathematics of elementary number theory: it locates the generalized Schatunowsky theorem in a very weak fragment of arithmetic, complements the earlier work of Pambuccian on Schatunowsky's theorem, and provides concrete model-theoretic independence proofs. The paper is explicit about the axioms used and gives meaningful independence results (A16 and A17 not derivable from the others, A18 not needed for GSs). The use of PA^- with explicit nonstandard models and the appeal to the Kaneko-Nakai classification are appropriate external anchors. The main obstacle is a formal flaw in the displayed axiom A17 that makes the stated theories inconsistent; the intended content is clear from the prose and from usage, but as printed the central theorems are vacuously true and the claimed proof is not sound.
major comments (3)
- [Section 3, A17] The displayed axiom A17 states (∀p)(∃q)(∀u)(π2(p)→(q<p∧π2(q)∧π2(u)∧u<p→u≤q)). With A18 instantiated at n=5, PA^- proves π2(2), since 2 is the only element whose square is <5. Applying A17 to p=2 forces existence of q with 1<q<2, but A14 (0<1 and x>0 implies x=1 or x>1) and A13 together rule out any such q in PA^-. Hence the theory Σ = A1-A19 is unsatisfiable, and Theorem 4.1's claim 'Σ⊢GSw' is vacuous. Theorem 4.2 inherits the problem because Σ' contains A17. The prose immediately after A17 says 'any prime > 2 has a predecessor prime', and the proof uses A17 only for primes ≥17, so the intended axiom should carry the hypothesis 2<p. This is a load-bearing error: as displayed, the paper does not prove the generalized Schatunowsky theorem from a satisfiable weak arithmetic. The fix is local (add 2<p to A17), and with that correction the nonstandard-model argument appears to go through, but the manuscript must be revised and the corrected axiom used consistently in Sections 4-6.
- [Theorem 4.1, paragraph after (4)] The proof that q^2 is a totative of m when q^2 and m are coprime uses the claim 'q^2 is co-prime with all the primes ≤ p' because q≥S(S(S(p))). This requires that all primes ≤p are below q, i.e., that the prime ordering is linear and that S is the true successor. In a nonstandard model of PA^- with A16 this is plausible, but the absence of induction means one must check that the finite set of primes ≤p is well-orderly; the manuscript does not spell out the argument. Since the proof otherwise depends on A19 for three consecutive primes, this is a minor gap in presentation rather than a fatal flaw, but it should be clarified.
- [Section 6, Theorem 6.1] The proof that C(Z[X]) satisfies GSs for nonstandard primes relies on the assertion that every axiom used in Theorem 4.2 holds in C(Z[X]), but earlier in Section 5 the paper shows only that A16, A17, A19, A20 hold in C(Z[X]) (and A21 is not discussed for this model). A21 is used in the definition of kp in Theorem 4.2, and the proof of Theorem 6.1 applies 'with kp defined as in that theorem'. The paper does not verify A21 in C(Z[X]) explicitly; while it may be derivable there, the missing verification is a gap in the independence claim 'PA^-, GSs ⊬ A18'.
minor comments (5)
- [Throughout] There are several typos and slips: 'costructive' for 'constructive' in the introduction; 'number-theroretical' for 'number-theoretical'; 'divsior' for 'divisor' in Theorem 4.2; 'Vi ete' for 'Viète'; 'succesor' for 'successor'; 'S(S(S(p)))' sometimes appears as 'S(S(S(p))' in the proof of Theorem 6.1. These should be corrected.
- [Section 3, A18] The axiom A18 states that for every n>4 there is a largest prime p with p^2<n. In the standard model this is true, but the formula as written uses 'p<q ∧ π2(q)→n≤q^2', which only says p is largest among primes greater than p; it does not explicitly say that all primes below p have square <n (which is automatic for p being the largest such prime). This is a minor clarity issue; the intended meaning is clear from context.
- [Section 4, GSw/GSs definitions] The formulas for GSw and GSs are dense and hard to parse. The paper should explain in words that GSw asserts an upper bound beyond which every number has a composite totative not divisible by any prime ≤p, and GSs asserts existence of a largest p-good number. This would help the reader follow the proof.
- [Section 5, Lemma 5.1] The statement of Lemma 5.1 assumes f(X) has degree n with coefficients a_i and constant term ±p; the proof refers to 'the polynomial having constant term ±1' after factorization, which is correct only up to sign of factors in Z[X]; a sentence clarifying the allowed factorization in Z[X] would improve rigor.
- [Reference [2]] The Kaneko-Nakai paper is cited as 'Amer. Math. Monthly 132 (2025), no. 5, 443-447'. The proof of Theorem 4.1 relies crucially on [2, Th. 4 & 7] and the table of p-good numbers; the manuscript would benefit from stating these classification results explicitly, or at least naming what the table contains, so the reader can verify the standard-model case without fetching the reference.
Circularity Check
No significant circularity: the proof derives GSw/GSs from explicit independent axioms and an external theorem; the displayed A17 inconsistency is a soundness bug, not a circularity.
full rationale
The claimed derivations are not circular. GSw and GSs appear only as target formulas, never as axioms or as assumptions. The added axioms A16-A21 are stated independently: successor and predecessor primes (A16, A17), largest prime below a square (A18), the three-consecutive-prime inequality (A19), Euclid VII.30 (A20), and the k_p inequality (A21). The proof of Theorem 4.1 uses A18 and A19 to construct a bound n = S(S(S(p)))^2 + 1 and to show that any larger m would have a composite totative coprime to all primes at most p; Theorem 4.2 adds A20 and A21 to make the largest p-good number explicit. The standard-model cases and the truth of A19 are imported from Kaneko-Nakai [2], an independent published result, not from the present paper or from the target statement. The only self-citation, [4], is a background remark about Bonse and Chebyshev inequalities and is not load-bearing. Section 6 also openly states that it is unknown whether all added axioms are needed, a limitation rather than a circularity. Separately, the displayed A17 appears formally unsatisfiable: A18 forces 2 to be prime, and A17 then demands a predecessor prime q with 1 < q < 2, contradicting PA^-'s order axioms; the surrounding prose ('any prime > 2') indicates a missing hypothesis. This is a correctness problem in the axiomatization as displayed, but it is not an equivalence between inputs and conclusions, so it does not raise the circularity score.
Assumptions & free parameters
assumptions (8)
- standard math A1-A15: PA^-, the 15 axioms of a discretely ordered commutative semiring without induction
- ad hoc to paper A16: every π2-prime has a successor prime
- ad hoc to paper A17: every prime has a predecessor prime (displayed version; prose says primes > 2)
- ad hoc to paper A18: for every n>4 there is a largest prime p with p^2<n
- ad hoc to paper A19: for consecutive primes r<p<q with q>17, q^2<2pr
- ad hoc to paper A20: every n>1 has a prime divisor (Euclid VII.30)
- ad hoc to paper A21: for consecutive primes p,q there is k with kp<q^2 and (k+1)p>q^2
- domain assumption Kaneko-Nakai classification of p-good numbers in the standard model [2, Th. 4 and 7]
Cite this review
Pith. "Pith review of Generalized Schatunowsky theorem in a weak arithmetic." pith.science (2026). https://pith.science/paper/4KMNBLH5
@misc{pith2026250608256,
author = {Pith},
title = {Pith review of: Generalized Schatunowsky theorem in a weak arithmetic},
year = {2026},
howpublished = {\url{https://pith.science/paper/4KMNBLH5}},
note = {Machine review of arXiv:2506.08256}
}
abstract
Schatunowsky's 1893 theorem, that 30 is the largest number all of whose totatives are primes, has been recently generalized by Kaneko and Nakai. In its generalized form, it states the finiteness of the set of all positive numbers $n$, which, for a fixed prime $p$, have the property that all of $n$'s totatives that are not divisible by any prime less than or equal to $p$ are prime numbers. It is this generalized form that we show holds in a weak arithmetic
Reference graph
Works this paper leans on
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[1]
L. E. Dickson, History of the theory of numbers, vol. I (Chelsea, 1952)
work page 1952
- [2]
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[3]
Kaye, Models of Peano Arithmetic (Oxford University Press, 1991)
R. Kaye, Models of Peano Arithmetic (Oxford University Press, 1991)
work page 1991
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[4]
V. Pambuccian, Schatunowsky’s theorem, Bonse’s inequality, and Chebyshev’s theorem in weak fragments of Peano arithmetic. Math. Log. Quart. 61, 230–235 (2015)
work page 2015
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[5]
Osada, The Galois groups of the polynomials X n + aX l + b, J
H. Osada, The Galois groups of the polynomials X n + aX l + b, J. Number Theory 25, 230–238 (1987). School of Mathematical and Natural Studies, Arizona State University - West V al- ley Campus, P. O. Box 37100, Phoenix AZ 85069-7100, U.S.A. Email address : hala.king@asu.edu; pamb@asu.edu
work page 1987
Reviewed August 7, 2026 · model on record in the stance chip above.
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