We develop the first case of Fermat's Last Theorem for exponent $p$ in the case where $p$ and $6p+1$ are prime (so that Sophie Germain's proof does not directly apply).

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We show that every integral $n$-by-$n$ matrix is the sum of four squares, for each $n\geq 2$.

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For any simple Eulerian graph of order $n$ and maximum degree $\leq 4$ we show that one needs $\leq (n-1)/2$ edge disjoint cycles to partition the edges of the graph.

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