Methods of Solving Number Theory Problems by Ellina Grigorieva

Author:Ellina Grigorieva
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Proof.

Assume contradiction and that Since n and m are both natural numbers and we must have n>m, we see that and and are both natural numbers. Since 2 is prime, it follows that and . Adding these two equations, we get , which means n is not a natural number. We obtain contradiction.

Therefore, 2 cannot be written as the difference of two squares.

However, we know that some even numbers, such as 44 or 100 mentioned above or other even numbers, can be written as difference of two squares:

can be written as difference of two squares.

Problem 150

Prove that any even number that can be written as a difference of two integer squares must be divisible by 4.

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