The sum of the product of four continuous natural numbers and 1 must be the complete square of a certain natural number 1) Calculation: (x + 1) (x + 2) (x + 3) (x + 4) 2) In the above formula, X is taken as 0.1.2.3... (1) what are the characteristics of the sum of the value of the polynomial and 1?

The sum of the product of four continuous natural numbers and 1 must be the complete square of a certain natural number 1) Calculation: (x + 1) (x + 2) (x + 3) (x + 4) 2) In the above formula, X is taken as 0.1.2.3... (1) what are the characteristics of the sum of the value of the polynomial and 1?

Let four continuous natural numbers be n-1, N, N + 1, N + 2
(n-1)n(n+1)(n+2)+1
=(n^2-1)(n^2+2n)+1
=n^4+2n^3-n^2-2n+1
=(n^2+n-1)^2
So the sum of the product of four continuous natural numbers and 1 must be a complete square number

Proof: the product of four continuous natural numbers is a complete square number

The following is a counterexample: 1 * 2 * 3 * 4 = 24 is not a perfect square number. It should be the product of four continuous natural numbers and the sum of 1 is a perfect square number. The proof is as follows: let these four continuous positive integers be: n, N + 1, n + 2, N + 3, (n > 0), then n (n + 1) (n + 2) (n + 3) + 1 = n (n + 3) (n + 1) (n + 2) + 1 = (n ^ 2 + 3n) (n ^ 2 + 3N + 2) +

The product of four continuous natural numbers is 11880. These four numbers are () methods

9, 10, 11, 12 I'm taking this number apart, 11880 = 2 * 2 * 2 * 3 * 3 * 3 * 5 * 11. Maybe I can get the above result with a try