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3a = x^2 + 2y^2 ==> a = u^2 + 2v^2

Source: Bundeswettbewerb Mathematik 2005, 1st Round, problem 2

March 1, 2005

calculusnumber theory proposednumber theory

Problem Statement

Let

a

be such an integer, that

3a

can be written in the form

x^2 + 2y^2

, with integers

x

and

y

. Prove that the number

a

can also be written in this form. Additional problems: a) Find a general (necessary and sufficent) criterion for an integer

n

to be of that form. b) In how many ways can the integer

n

be represented in that way?