Question 1059605
Let the four consecutive integers be defined based on their avetshe, {{{a}}} , as
{{{a-3/2}}} , {{{a-1/2}}} , {{{a+2/2}}} , and {{{a+3/2}}} .
The sum the problem talks about is
{{{(a-3/2)(a-1/2)(a+1/2)(a+3/2)+1}}}=
{{{(a^2-9/4)(a^2-1/4)+1}}}=
{{{a^4-(10/4)a^2+9/16+1}}}=
{{{a^4-2(5/4)a^2+25/16}}}=
{{{(a^2-5/4)^2}}} .
{{{a^2-5/4}}} is an integer.
In fact, it is
the sum of {{{1}}} plus the product if the first and fourth integers:
{{{1+(a-3/2)(a+3/2)=4/4+a^2-9/4=a^2-5/4}}} .