, the proof goes through the construction of a sequence of integers by concatenating blocks whose largest prime factor is constant, and linking blocks together with separating integers
, For an integer m ? 2, P ? (m) denotes the smallest prime factor of m. Let N ? 1. A path of integers ? N of length l is a l-uple C =, IV.2 Notation The letters p, q, q , r, r will always denote generic prime numbers
, N } into paths is said to be optimal if it contains F (N ) paths (see the Introduction for the definition of F )
Then C is said to be n-factorable if all the integers of C are multiple of n. Then C can be written as C = nD where D is a path of integers ? N/n. For integers 1 ? n ? N ,
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