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Statement

For $n\ge1$ let $a_1<a_2<\cdots<a_{\varphi(n)}$ be the integers in $[1,n]$ relatively prime to $n$ (for $n\ge2$, those in $[1,n-1]$). The paper does not restate this definition for general $n$; it carries over the notation it has just used for the product $n_k$ of the first $k$ primes (p. 80).

Conjecture (13) (p. 80), as posed: "There is an absolute constant $c$ so that for every $n$

$$\sum_{i=1}^{\varphi(n)-1}(a_{i+1}-a_i)^2<\frac{cn^2}{\varphi(n)}$$

"

Erdős calls it one of his favourite conjectures, about 45 years old, says that Hooley did significant work on it but that it is still open, and offers a prize for a proof or disproof (p. 80).

Prime analogue (14) (p. 80):

$$\sum_{p_k<x}(p_{k+1}-p_k)^2<cx\log x,$$

which he calls "completely out of reach" (p. 80). The quantifier on $c$ in (14) is not restated; it is read as for (13).

Source. P. Erdős, On some of my problems in number theory I would most like to see solved, Number Theory (Ootacamund, 1984), Lecture Notes in Mathematics 1122, Springer, 1985, 74--84; (13) and (14) on p. 80. The edition is identified on the source card.

Read depth. Claims checked: (13), (14) and the surrounding remarks were read clause by clause on the page image.

Proof pointer

None; the paper states (13) and (14) as conjectures.

Dependencies

None.

Bears on

• Problem 220: the problem asks whether $\sum_{1\le k<\varphi(n)}(a_{k+1}-a_k)^2\ll n^2/\varphi(n)$ for the integers coprime to $n$, which is (13) as posed. The paper records it as open in 1985 and proves nothing on it.