Erdos 1937 sum difference squares primes

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theorem_section_1: Erdős's theorem that for infinitely many n the equation n = p^2 + q^2 in primes p, q has more than n^(c_3/log log n) solutions, improving the n^(c_2/(log log n)^2) of Part I.

theorem_section_2: Erdős's theorem that if an infinite increasing sequence of positive integers has more than N^(1 - c_4/log log N) terms up to N for infinitely many N, with c_4 below half of log 2, then for infinitely many M the equation r_j^2 - r_i^2 = M has more than M^(c_5/log log M) solutions.

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P. Erdős: On the sum and difference of squares of primes (II), J. London Math. Soc. 12 (1937), 168--171; Zentralblatt 17,103.

This sequel to Part I (J. London Math. Soc. 12 (1937), 133-136) states two theorems in its introduction (p. 168) and proves them in Sections 1 and 2. Section 1 (pp. 168-170) proves that for infinitely many n the equation n = p^2 + q^2 with p, q prime has more than n^{c_3/log log n} solutions, improving the n^{c_2/(log log n)^2} bound of Part I; the paper says the principal difference from Part I, whose proofs were elementary, is that the argument requires Brun's method, which enters through the Brun-Titchmarsh upper bound for primes in arithmetic progressions in the proof of the lemma. The construction takes A = 513...*p_k, the product of the first k primes of the form 4d+1, factors it as A = a_1...a_x with x a sufficiently large absolute constant and each a_i having at least [k/x] prime factors, and uses a lemma (p. 168) that some a_i has more than A^2/(phi(a_i)(log A)^2) primes p < A^2 in each of at least (7/8)phi(a_i) residue classes mod a_i. Section 2 (pp. 170-171) generalizes the result proved in Section 1 of Part I: if r_1 < r_2 < ... is an infinite sequence of positive integers with, for infinitely many N, more than N^{1 - c_4/log log N} terms up to N, where c_4 < (1/2) log 2, then for infinitely many M the equation r_j^2 - r_i^2 = M has more than M^{c_5/log log M} solutions, with c_5 depending only on c_4.

Source: https://users.renyi.hu/~p_erdos/1937-08.pdf. No notice is printed on the offprint scan, which reads "[Extracted from the Journal of the London Mathematical Society, Vol. 12, 1937.]", and the hosting archive's index states no terms (https://users.renyi.hu/~p_erdos/Erdos.html, read 2026-10-02); the publisher's page for this article was not consulted, Wiley's page for a 1936 article in the journal (DOI 10.1112/jlms/s1-11.2.133) could not be read on 2026-10-02, and that article's Crossref record lists the version-of-record license http://onlinelibrary.wiley.com/termsAndConditions#vor, whose Wiley Online Library Terms and Conditions (archived capture of 2024) state "As a User, you have certain rights specified below; all other rights are reserved."; the London Mathematical Society's journal page describes the journal as "Hybrid open access" with rights and permissions handled by Wiley (https://www.lms.ac.uk/publications/jlms, read 2026-10-02), every other right reserved.

Bears on. #979: the problem asks whether the number f_k(n) of representations of n as a sum of k kth powers of primes has limsup infinity for every k at least 2; the Section 1 theorem gives, for k = 2, more than n^{c_3/log log n} representations for infinitely many n, so f_2 is unbounded, as Part I had already shown with a weaker bound; the paper says nothing about k at least 3.

Results. the sum theorem (stated p. 168, unnumbered; proved in Section 1, pp. 168-170); the difference theorem (stated p. 168, unnumbered; proved in Section 2, pp. 170-171). The Lemma (p. 168) is a proof step of the sum theorem, stated on its page.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.