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Statement

Setting (p. 103). $u_1=1<u_2<\cdots$ are the integers of the form $x^2+y^2$, and $c_1,c_2,\ldots$ are absolute constants.

Inequality (2) (p. 103). For infinitely many $i$,

$$u_{i+1}-u_i>c_2\,\frac{\log u_i}{(\log\log u_i)^{1/2}}.\qquad(2)$$

Context on p. 103. The paper records that Chowla and Bambah remarked $u_{i+1}-u_i<c_1u_i^{1/4}$, a bound whose proof the paper calls immediate; that the conjecture $u_{i+1}-u_i=o(u_i^{1/4})$, that is, for every $\varepsilon>0$ and every sufficiently large $n$ some integer of the form $x^2+y^2$ lies in $(n,n+\varepsilon n^{1/4})$, is still unproved; and that Turán observed, in a letter, the bound (1), $u_{i+1}-u_i>c_2\log u_i/\log\log u_i$ for infinitely many $i$, and asked whether it could be improved. Inequality (2) is Erdős's improvement. The print writes the constant of (1) as $c_2$, the same symbol as in (2).

Source. P. Erdős, Some problems and results in elementary number theory, Publ. Math. Debrecen 2 (1951), 103--109, doi:10.5486/pmd.1951.2.2.04: (2) on p. 103, its deduction from Theorem 1 on p. 104, and a second route on p. 106. The edition read is identified on the source card.

Read depth. Claims checked: the statement and the deduction were read clause by clause on the printed pages. Nothing here is independently reviewed.

Proof pointer

Page 104. Take the $p_i$ of Theorem 1 to be the primes $\equiv3\pmod4$. Every $u$ is then a $v$ (each such prime divides a sum of two squares to an even power), and $\sum_{p\equiv3\,(4),\,p\le x}1/p=\tfrac12\log\log x+O(1)$ gives $e^{f(\log v_i)}>c_5(\log\log v_i)^{1/2}$, so (3) yields (2). Page 106 notes that (2) can also be had without Brun's method, from Landau's count $At/(\log t)^{1/2}+o(t/(\log t)^{1/2})$ of the integers up to $t$ of the form $u^2+v^2$.

Dependencies

Theorem 1 of the same paper. The upper bound recorded above is Bambah and Chowla's, carded at bambah_1947_numbers_which_can_be_expressed_as.

Bears on

• Problem 222: (2) is a lower bound for infinitely many of the gaps $n_{k+1}-n_k$ the problem asks to bound. The paper gives no upper bound of its own; it records Bambah and Chowla's $c_1u_i^{1/4}$ and the unproved conjecture $o(u_i^{1/4})$.