Do they satisfy? (Part 3)

Level pending

See Part 2 and Part 1 if you haven't.

$P$ : sum of $2$ nonnegative squares.

$21$
$2017$
$2011^3 \cdot 7^{12}$
$2^{34857433} - 1$
$32493290$
$5130274$
$52554541$
$93390422178601$

The answer is 2005678.

1 solution

Chandler West
Feb 16, 2014

It can be shown that a number cannot be written as the sum of two positive squares if it has a prime factor equal to 3 (mod 4) with an odd power. All of the above numbers except for 4 can easily be factored by a computer.

For #4, test primes equal to 3 mod 4 until we find one that divides. The first is 7, and we can show that 7^1 divides but 7^2 does not. Therefore #4 cannot be written as the sum of two squares.

I'm probably being slow, but how do you show that 7 divides #4, but 49 does not?

Steven Perkins - 7 years, 3 months ago

Because it is not 3 mod 4.

Joshua Ong - 7 years, 2 months ago

The author should write that the nonnegative squares are integers and that their square roots are as well.

Sharky Kesa - 7 years, 3 months ago

Do they satisfy? (Part 3)

The answer is 2005678.

1 solution

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