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Not a Prime (Posted on 2012-12-20) Difficulty: 3 of 5
Prove that 5100+575+550+525+1 is not a prime number.

No Solution Yet Submitted by Danish Ahmed Khan    
Rating: 3.0000 (2 votes)

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Solution Try this | Comment 5 of 6 |

The idea is to complete squares.

Find a likely candidate, e.g. from here: A026374: 1,6,11,6,1.

Then (x^4+6x^3+11x^2+6x+1) - y = x^4+x^3+x^2+x+1

and y = (5x^3+10x^2+5x)

(x^4+6x^3+11x^2+6x+1) - (5x^3+10x^2+5x)=x^4+x^3+x^2+x+1

But (x^4+6x^3+11x^2+6x+1) = (x^2+3x+1)^2

And (5x^3+10x^2+5x) = 5x(x+1)^2

Now since x=5^25, 5x=x^26, in which case (5^13)^2(x+1)^2, completes a difference of squares that can be factored:

[(x^2+3x+1)-(5^13)(x+1)]*[(x^2+3x+1)+(5^13)(x+1)]

Very pretty.

Edited on December 21, 2012, 7:35 am
  Posted by broll on 2012-12-21 07:04:27

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