2&1

Number Theory Level 2

$2 \times 2 \times2 \times2 \times2 \times2 \times2 \times2 \times2\times 2\times 2\times 2\times 2\times 2\times 2\times 2 -1 = 65535$

What is its biggest prime factor?

3 15 17 257

2 solutions

James Pohadi
Mar 30, 2015

$2^{16}-1$ $=(2^8+1)(2^8-1)$ $=(2^8+1)(2^{8}-1)$ $=(2^8+1)(2^{4}+1)(2^{4}-1)$ $=(2^8+1)(2^{4}+1)(2^{2}+1)(2^{2}-1)$ $=(2^8+1)(2^{4}+1)(2^{2}+1)(2+1)(2-1)$

Since $(2^8+1)=257$ is a prime factor, then $\boxed{257}$ is the biggest prime factor.

Bryan Lee Shi Yang
Mar 3, 2015

2^16-1=255 x 257. Since 257 is a prime, then the largest factor must be 257.

2&1

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