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Statement

Setting (p. 107). $s_1<s_2<\cdots$ are the squarefree numbers. They are the sequence $v$ of Theorem 1 with the $p$'s taken to be the squares of the primes instead of primes, a case with $\sum1/p_i<\infty$.

Inequality (20) (p. 107). Erdős states that the method of Theorem 1 gives, for infinitely many $i$,

$$s_{i+1}-s_i>(1+o(1))\,\frac{\pi^2}{6}\,\frac{\log s_i}{\log\log s_i}.\qquad(20)$$

The print's (20) has $\pi^3/6$ [sic]. The next sentence speaks of replacing $\pi^2/6$ by a larger constant, and (21) carries $\pi^2/6$, so the constant of (20) is read here as $\pi^2/6$.

Context on p. 107. Erdős does not know whether (20) had been published, and says it was known to Bateman, Chowla and Mirsky, among others. He says it seems extremely hard to replace $\pi^2/6$ by a larger constant and says it seems possible that for $i>i_0$

$$s_{i+1}-s_i<(1+\varepsilon)\,\frac{\pi^2}{6}\,\frac{\log s_i}{\log\log s_i},\qquad(21)$$

which the paper poses without proof. The strongest result in the direction of (21) that it records is Roth's,

$$s_{i+1}-s_i<s_i^{3/13}(\log s_i)^{\frac4{13}+\varepsilon},\qquad(22)$$

from K. F. Roth, On the gaps between squarefree numbers, J. London Math. Soc. 26 (1951), 263--268 (footnote 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: displays (20)--(22) on p. 107. The edition read is identified on the source card.

Read depth. Claims checked: (20)--(22) and their framing were read clause by clause on the printed page. The paper gives no separate proof of (20), so none was checked.

Proof pointer

None written out. The paper says only that "our above method", the Chinese-remainder construction of Theorem 1, gives (20).

Dependencies

The construction in the proof of Theorem 1.

Bears on

• Problem 208: the problem's second question is the upper bound (21), posed here as seeming possible, and (20) shows that its constant $\pi^2/6$ could not be lowered. Roth's (22), recorded here, bounds the gaps by $s_i^{3/13+o(1)}$, which does not reach the problem's first question, a bound $\ll_\epsilon s_n^\epsilon$ for every $\epsilon>0$. The paper answers neither question.