It is proved that the product of k consecutive positive integers is not a complete square number

It is proved that the product of k consecutive positive integers is not a complete square number

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The proof of k = 101
Suppose that the product of 101 consecutive positive integers is a complete square number, then among the 101 positive integers, there are at most 2 multiples of 97, 2 multiples of 89, 2 multiples of 83, 2 multiples of 79, 2 multiples of 73, 2 multiples of 71, 2 multiples of 67, 2 multiples of 61, 2 multiples of 59, 2 multiples of 53, 3 multiples of 47, 3 multiples of 43, 3 multiples of 41, 3 multiples of 37, 4 multiples of 31, 4 multiples of 29, Five multiples of 23, six multiples of 19, six multiples of 17, eight multiples of 13, ten multiples of 11, and fifteen multiples of 7
101-2*10-3*4-4*2-5-6*2-8-10-15=11
So the prime factor of at least 11 numbers is either 2, 3, 5 or not less than 101
Obviously, the times of prime factors of these numbers are even
If the times of 2,3,5 of these numbers are ≥ 2, then an even number can always be subtracted to make the times 0 or 1
So these 11 numbers can be expressed in the following form: (2 ^ RI) * (3 ^ SI) * (5 ^ Ti) * (UI ^ 2), where RI, Si and Ti are 0 or 1, UI is a positive integer, 1 ≤ I ≤ 11
Because there are only 8 groups of different arrays (RI, Si, Ti), according to the drawer principle, there must be two arrays that are exactly the same. Then the product of the two corresponding numbers is a complete square number
Let these two numbers be m, M + N, then M (M + n) is a complete square number, where m and N are positive integers and 1 ≤ n ≤ 100
Let (m, n) = D, then M = DS ^ 2, M + n = DT ^ 2, where D, s, t are positive integers, s2ds
So m = DS ^ 2 ≤ (DS) ^ 2 ≤ 49 ^ 2 = 2401
This shows that one of the 101 consecutive positive integers must not exceed 2401
According to the 26 prime numbers 101199293389487683773863953105111511249132714271523162117211811190720992179227323712467, we can know that as long as one of the 101 consecutive positive integers does not exceed 2401, there must be one of the 26 prime numbers, which obviously contradicts the fact that the product of the 101 consecutive positive integers is a complete square number