SOLUTION: Prove that if 1 is added to the product of any four consecutive integers, the sum is a perfect square. Thank you in advance.

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Question 1059605: Prove that if 1 is added to the product of any four
consecutive integers, the sum is a perfect square.
Thank you in advance.
Found 3 solutions by ikleyn, Boreal, KMST:
Answer by ikleyn(52794)   (Show Source): You can put this solution on YOUR website!
.
Prove that if 1 is added to the product of any four
consecutive integers, the sum is a perfect square.
Thank you in advance.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

The problem asks us to prove that n*(n+1)*(n+2)*(n+3) + 1 is a square of an integer.


Let x =  be the central point for the original four integers n, n+1, n+2 and n+3.  Then

  n*(n+1)*(n+2)*(n+3) + 1 = (x+0.5)*(x-0.5)*(x+1.5)*(x-1.5) + 1 = 

=  =  = 

=  = .


Next,   = (by the definition of "x") =  =   is the integer number.


Thus we proved that  n*(n+1)*(n+2)*(n+3) + 1  is the square of the integer number   = .

Proved and solved.


Answer by Boreal(15235)   (Show Source): You can put this solution on YOUR website!
x(x+1)(x+2)(x+3)+1=x(x+3)(x+2)(x+1)+1
so (x^2+3x)(x^2+3x+2)+1=(x^2+3x+1)(x^2+3x+2)-(x^2+3x+2)+1
You are converting the first term to (x^2+3x+1), and that is adding x^2+3x+2 to that side. That means you have to subtract it.
Convert the (x^2+3x+2) to (x^2+3x+1). But that requires adding the factor x^2+3x+1. because by subtracting one from that term, which is multiplied by (x^2+3x+1), you have to add it back.
Then you have (x^2+3x+1)(x^2+3x+1)-(x^2+3x+2)+(x^2+3x+1)+1. The last two terms disappear because distributing the minus sign give -x^2-3x-2+x^2+3x+1+1.
You are left with (x^2+3x+1)^2, which is a perfect square.
Pick an x, like 7
7,8,9,10 has a product of 5040 and 5041, 1 more, is a perfect square of
x^2+3x+1, or 49+21+1, or 71.

Answer by KMST(5328)   (Show Source): You can put this solution on YOUR website!
Let the four consecutive integers be defined based on their avetshe, , as
, , , and .
The sum the problem talks about is
=
=
=
=
.
is an integer.
In fact, it is
the sum of plus the product if the first and fourth integers:
.
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