# Math Help - number theory

1. ## number theory

evaluate legender symbol

( 2 / 2^43112609 - 1)

ie ( 2/largest prime)

2. There is no largest prime.

Use the fact that $(2/p)=1$ for $p\equiv\pm 1 \mod 8$ and $(2/p)=-1$ for $p\equiv\pm 3 \mod 8$.

3. Out of curiosity, how would you evaluate linear congruences like
$2^{43112609} - 1 \equiv a \mbox{ (mod 8)}$if you have BIG numbers, without a calculator or computer?

4. If $a \equiv b$ (mod c), then $a^k \equiv b^k$ (mod c), I guess ?

5. Originally Posted by ilikecandy
Out of curiosity, how would you evaluate linear congruences like
$2^{43112609} - 1 \equiv a \mbox{ (mod 8)}$if you have BIG numbers, without a calculator or computer?
Think about it : $2^n \equiv 0 \mod 8$ for $n \geq 3$. So $2^{43112609} - 1 \equiv -1 \mod 8$.