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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