Prove that 9n can be expressed in the required form...
Source: INMO 2007 Question 2
November 2, 2009
number theory unsolvednumber theory
Problem Statement
Let be a natural number such that n \equal{} a^2 \plus{} b^2 \plus{}c^2 for some natural numbers . Prove that
9n \equal{} (p_1a\plus{}q_1b\plus{}r_1c)^2 \plus{} (p_2a\plus{}q_2b\plus{}r_2c)^2 \plus{} (p_3a\plus{}q_3b\plus{}r_3c)^2
where 's , 's , 's are all nonzero integers. Further, if does not divide at least one of prove that can be expressed in the form x^2\plus{}y^2\plus{}z^2, where are natural numbers none of which is divisible by .