Number Theory

Recall the identity $x^5+1 = (x+1)(x^4-x^3+x^2-x+1)$ Substitute $x = 10^{20}$ will give $10^{100}+1 = (10^{20}+1)(10^{80}-10^{60}+10^{40}-10^{20}+1)$ Hence proved, this number is NOT a prime.

No, googol+1 isn't prime and you don't need a computer to see that.

If mm is odd, then

(−1)m=−1(−1)m=−1

so

(−1)m+1=0(−1)m+1=0

or put differently, −1−1 is a root of the polynomial Xm+1Xm+1 , which is the same as saying that the polynomial Xm+1Xm+1 is divisible by the polynomial X+1X+1 .

This means that whatever positive integer you substitute for XX in Xm+1Xm+1 when mm is odd, the result will be divisible by X+1X+1 . So, Xm+1Xm+1 cannot be prime (unless m=1m=1 or you've substituted 1 for XX , in which case the divisor we found, X+1X+1 , is the same as the number Xm+1Xm+1 ).

This is why 25+1=3325+1=33 is divisible by 2+1=32+1=3 .

This is why 103+1=1001103+1=1001 is divisible by 10+1=1110+1=11 .

This is why 7777+17777+1 is divisible by 77+1=7877+1=78 . I don't even need to calculate that number; I already know it's divisible by 7878 .

So, all we need to do is take m=25m=25 (which is odd) and X=104X=104 to find that

(104)25+1(104)25+1 is divisible by 104+1104+1 .

So googol+1 is divisible by 10,001. It is not a prime number. $LaTeX$

Prime number is of the form 2^n-1