Too large to calculate

Number Theory Level 3

How many distinct prime factors does the following number have:

$2 ^ { 32 } + 2 ^{ 17 } + 1 ?$

1 8 4 2

6 solutions

Isaac Buckley
Aug 20, 2015

$\LARGE 2^{32}+2^{17}+1=(2^{16}+1)^2$

And $2^{16}+1$ is a Fermat prime so the answer is $\boxed{1}$ .

Moderator note:

Great use identifying the Fermat's prime.

Fun fact: $2^{16} + 1 =65537$ is the largest known Fermat's prime.

Sadasiva Panicker
Aug 26, 2015

2^32+2^7+1 = (2^16+1)^2 = 65537^2: 65537 is a prime, So 1 prime number

Pranay Kumar
Aug 24, 2015

It's a perfect square of ((2^16) +(1)).. which is a fermet prime number

Soldà Federico
Aug 21, 2015

$2^{32}+2^{17}+1=(2^{16})^2+2*(2^{16})*1+1^{2}=(2^{16}+1)^{2}$ Since the number is a perfect square, the exponents of the decomposition into prime factors are even, and then the number of divisors must be the product of odd numbers, and then an odd number.

You still need to show that $2^{16} + 1$ is prime.

Chung Kevin - 5 years, 9 months ago

how 2^16+1 will be prime number?

Khyati Mirani - 5 years, 9 months ago

You don't know if $2^{16}+1$ is a prime number, but all perfect square numbers have an odd number of divisors and 1 is the only odd posibility.

Soldà Federico - 5 years, 9 months ago

$2^{16}$ is a perfect square, but $2^{16} + 1$ is not a perfect square.

Chung Kevin - 5 years, 9 months ago

@Chung Kevin – Sorry, $(2^{16}+1)^{2}$ is a perfect square.

Soldà Federico - 5 years, 9 months ago

@Soldà Federico – Do you know why it is perfect square ?

Syed Baqir - 5 years, 9 months ago

It is Fermat prime

Syed Baqir - 5 years, 9 months ago

Jason Chrysoprase
Sep 12, 2015

Actually, 1 is not a prime number

Hadia Qadir
Aug 30, 2015

4295098369 is 2^32+2^17+1 from what I can tell is a prime number and that is 1

No, that number is not a prime.

Chung Kevin - 5 years, 9 months ago

Too large to calculate

6 solutions

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