Problem
NT-B2-M02-P008 Three Squares
#8
★★★☆☆ Level 3 of 5
Prove that \(x^2+y^2+z^2=8t+7\) has no integer solutions.
Add three possible residues from the set \(\{0,1,4\}\).
Modulo \(8\), each square is \(0,1\), or \(4\). Checking sums of three such residues shows that residue \(7\) is impossible: the possible residues include \(0,1,2,3,4,5,6\), but not \(7\). The right-hand side has residue \(7\), contradiction.
This prepares for classical three-square problems without stating the full theorem.