2

Part of 2005 Bundeswettbewerb Mathematik

Problems(2)

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

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

a

be such an integer, that

3a

can be written in the form

x^2 + 2y^2

, with integers

x

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?

calculusnumber theory proposednumber theory

Finite number of triples with given properties

Source: German Mathematical Competition BWM 2005, 2nd round, problem

x

a rational number. Prove: There are only finitely many triples

(a,b,c)

of integers with

a<0

b^2-4ac=5

ax^2+bx+c

algebra proposedalgebra