The following theorems are usually considered so elementary by professional mathematicians that they are stated without proof. For hobbyists like me, to whom they may not be so transparent, proofs are provided in the attached pdf. In what follows (1) ind_{p}n represents the index of the prime p in the prime composition of n – for example ind_{2} 24 = 3, ind_{3} 24 = 1, ind_{5} 24 = 0 (2) [n] is the largest integer less than or equal to n and (3) L_{n} is the least positive integer divisible by each of 1, 2, … , n.

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