REVIEW
An additive problem in connection with some quadratic real fields with class number one
Number Theory · Submitted 15 Sep 2026 · 10 references
Pith's verdict: Competent, narrow classification paper: the p≡1 case is genuinely new, the p≡3 stress-test worry is a false alarm, but Section 5 needs referee arithmetic.
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Referee report
The paper
The paper claims
All primes p such that p minus an odd square has at most two prime factors are classified, with one possible exception.
The reading
What carries the argument
The proof relies on four separate strategies corresponding to each residue class mod 8. For p ≡ 7 (mod 8) the author shows p = $a^{2}$ − 2 and then uses a theorem of Jungyun Lee on Richaud-Degert types to bound the parameter. For p ≡ 5 (mod 8) the problem reduces to cases p = $m^{2}$ + 4 or p = 1 + $4n^{2}$, and the class number one property of Q(√p) is established via Biró's proofs of Yokoi's conjecture and Chowla's conjecture. For p ≡ 3 (mod 8) the argument splits into subcases; one subcase uses Lee's theorem and the other relies on the Mollin–Williams classification of real quadratic fields of class number one with continued fraction period length six. For p ≡ 1 (mod 8) the method is elementary and uses a lemma that forces p to be a quadratic residue modulo at least one prime in a specific set, leading to an explicit bound p < 16·433², after which a direct search yields all primes.
Extended reading notes
Core claim
Theorem 1.1 gives the complete list of primes p satisfying the condition ω(p − $x^{2}$) ≤ 2 for every odd x with $x^{2}$ < p. For p ≡ 7 (mod 8) the primes are 7, 23, 47, 167. For p ≡ 5 (mod 8) they are 5, 13, 29, 37, 53, 101, 173, 197, 293, 677. For p ≡ 1 (mod 8) they are 17, 41, 73, 89, 97, 113, 137, 233, 257, 353, 593, 857, 1097, 1217, 1553, 2273, 5297. For p ≡ 3 (mod 8) they are 3, 11, 19, 59, 83, 107, 227, with one possible exception that is excluded only under the Riemann Hypothesis.
Load-bearing premise
The proof for p ≡ 3 (mod 8) depends on the Mollin–Williams theorem which states that at most 19 squarefree d have class number one and continued fraction period length six, with only 18 known; the final exclusion of the possible exception in that case requires the Riemann Hypothesis.
Editorial extensions
If this is right
- Any prime not appearing in the four lists of Theorem 1.1 does not satisfy the ω(p − x^2) ≤ 2 condition for all odd x < √p.
- The classification provides a finite and completely known set, except possibly one undetermined prime in the p ≡ 3 (mod 8) case.
- The result confirms that the condition forces very strong arithmetic restrictions on p, in particular forcing Q(√p) to have class number one.
- The elementary method for p ≡ 1 (mod 8) shows that heavy computational search can be avoided by a clever combination of quadratic residues and bounds.
- The one possible exception for p ≡ 3 (mod 8) is linked to the existence of an unknown real quadratic field of class number one with continued fraction period length six.
Reading between the lines
- The condition ω(p − x^2) ≤ 2 for all odd x < √p is so restrictive that only finitely many primes satisfy it, and the author has essentially found them all. This is reminiscent of the Rabinowitsch criterion for imaginary quadratic fields and suggests a deeper connection between additive constraints on prime gaps and class number one.
- The elementary approach for p ≡ 1 (mod 8) demonstrates that sophisticated analytic machinery is not always necessary; a systematic quadratic residue chase can suffice for complete classification.
- If the Riemann Hypothesis is false, the possible exceptional prime for p ≡ 3 (mod 8) might actually exist, so the list in that case would be conditional. Conversely, if the Riemann Hypothesis is true, the list is unconditional and complete.
- The results could be used as a benchmark for testing computational number theory packages: verifying that no other prime below any reasonable bound satisfies the condition would confirm the theorem's completeness in practice.
Original abstract
Our aim is to find all the prime numbers $p$ such that $p-x^2$ has at most two different prime factors, for all the odd integers $x$ such that $x^2<p$. We solve entirely the cases $p\equiv 1,5,7 \pmod 8$. The case $p\equiv 3 \pmod 8$ is solved with one possible exception.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
Our aim is to find all the prime numbers $p$ such that $p-x^2$ has at most two different prime factors, for all the odd integers $x$ such that $x^2<p$. We solve entirely the cases $p\equiv 1,5,7 \pmod 8$. The case $p\equiv 3 \pmod 8$ is solved with one possible exception.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Our aim is to find all the prime numbers $p$ such that $p-x^2$ has at most two different prime factors, for all the odd integers $x$ such that $x^2<p$. | contribution 0.67 | 0.48 | 0.09 | asserted 0.88 |
| s1 | We solve entirely the cases $p\equiv 1,5,7 \pmod 8$. | result 0.81 | 0.95 | 0.31 | asserted 1.00 |
| s2 | The case $p\equiv 3 \pmod 8$ is solved with one possible exception. | result 0.71 | 0.94 | 0.18 | asserted 0.78 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
References
Discussion
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