Consider the \(4!\) possible permutations of the numbers \(1,2,3,4\). Which of those permutations keep the expression \(x_1x_2+x_3x_4\) the same?
Show that if \(1+3+5+7+...+97+99=50^2\), then \(1+3+5+7+...+97+99+101=51^2\).
Prove that for all positive integers \(n\) there exists a partition of the set of positive integers \(k\le2^{n+1}\) into sets \(A\) and \(B\) such that \[\sum_{x\in A}x^i=\sum_{x\in B}x^i\] for all integers \(0\le i\le n\).
Diophantine equations are those where we’re only interested in finding the integer solutions. Some of these equations are quite simple, while others look simple but are actually very difficult. The most famous one is Fermat’s Last Theorem, which says that when \(n>2\), there are no solutions to \[x^n+y^n=z^n.\] Pierre de Fermat claimed that he proved this in 1637, scribbling it in the margin of a book, but said “I have discovered a truly marvelous proof of this, which this margin is too narrow to contain." It was only proved by Andrew Wiles in 1995. Today’s problems won’t take 358 years to solve.