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Statement

In the notation of displays (5) and (6) of p. 27 (see the p. 27 conjecture), the exponents (6) are those of all primes in the factorization of $\prod_{i=1}^{n}(x+i)$. On pp. 27--28 Erdős reports that several years earlier he and Selfridge proved that the product (5) is never a power, and states their conjecture:

Conjecture (Erdős and Selfridge, p. 28). At least one of the exponents (6) equals $1$.

Erdős adds that the conjecture seemed hopeless to them then, "still is hopeless and probably will stay so for some time" (p. 28). The paper gives no proof or evidence for it.

The print states no range for $n$ or $x$. For $n=2$ the conjecture fails as stated: $x=7$ gives $8\cdot9=2^3\cdot3^2$, in which no exponent is $1$ (an observation of this page, not of the paper). The conjecture is therefore read for $n\ge3$, the range of Problem 137.

Source. P. Erdős, Miscellaneous problems in number theory, Proceedings of the Eleventh Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, Man., 1981), Congr. Numer. 34 (1982), 25--45; the report and the conjecture on pp. 27--28. The never-a-power theorem is cited in the paper's list on p. 37 as P. Erdős and J. L. Selfridge, The product of consecutive integers is never a power, Illinois J. Math. 19 (1975), 292--301. The edition read is identified on the source card.

Read depth. Claims checked: the passage was read clause by clause on the page images. The 1975 theorem is reported here, not proved; this page does not check it.

Dependencies

The Erdős--Selfridge theorem of 1975, cited above. The stronger conjecture of p. 27 asks for at least $k$ exponents equal to $1$ once $n$ is large.

Bears on

• Problem 137: for $n\ge3$ the conjecture says that a product of $n$ consecutive positive integers always has a prime dividing it exactly once, which is the negative answer to the problem's question whether such a product can be powerful. The paper poses it and records no result on it beyond the never-a-power theorem, which concerns perfect powers rather than powerful numbers.