Factors of consecutive products

Pick a positive integer k and take the product of k consecutive integers greater than k. Then the result is divisible by a prime number greater than k. This theorem was first proved 128 years ago [1].

For example, suppose we pick k = 5. If we take the product

20*21*22*23*24

then it’s obviously divisible by a prime larger than 5 because 23 is one of the factors. But we could also take, for example,

32*33*34*35*36.

Although the sequence {32, 33, 34, 35, 36} doesn’t contain any primes, it contains a couple numbers with prime factors larger than 5, namely 33 = 3*11 and 34 = 2*17. Sylvester’s theorem guarantees that there will always be one number out of 5 consecutive integers that is divisible by a prime greater than 5, provided the first number is itself greater than 5.

For one more example, we can pick k = 8 and look at

140*141*142*…*147.

One of the numbers between 140 and 147 (inclusive) must be divisible by a prime greater than 8, and in fact five of them are, starting with 141 = 3*47.

In both of our examples there were multiple numbers that had a prime factor large enough to satisfy Sylvester’s theorem. Are there examples where only one number has a factor large enough? Yes, but not many.

For example, {8, 9} is a pair of consecutive integers, only one of which has a prime factor bigger than 2. And {8, 9, 10} is an example of 3 consecutive integers, only one of which has a prime factor larger than 3. I wrote a program to search for examples with k = 4 and didn’t find any.

In any case, a theorem dating back to 1897 [2] proves that there could only be finitely many examples for k > 2. Lehmer extends the results [2] in his paper [3].

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[1] J. J. Sylvester, On arithmetical series, Messenger Math., 21 (1892) 1-19, and 87-120; Collected Mathematical Papers, 4 (1912) 687-731.

[2] G. St0rmer, Quelques theoremes sur l’equation de Pell x² – Dy² = ± 1 et leurs applications,
Skr. Norske Vid. Akad. Oslo, I no. 2 (1897).

[3] D. H. Lehmer, The Prime Factors of Consecutive Integers, The American Mathematical Monthly , Feb., 1965, Vol. 72, No. 2, Part 2, pp. 19-20.

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