MathDB

Let

S

be the set of all positive integers

n

such that

n^4

has a divisor in the range

n^2 +1, n^2 + 2,...,n^2 + 2n

. Prove that there are infinitely many elements of

S

of each of the forms

7m, 7m+1, 7m+2, 7m+5, 7m+6

and no elements of

S

of the form

7m+3

and

7m+4

, where

m

is an integer.

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