By Waclaw Sierpinski, I. N. Sneddon, M. Stark

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C) are primes and we may suppose that where p',p",pf", p' ύ ρ" ύ ρ'" ύ ··· ^ P{k) (as for each of the numbers n, ri,... we denote thereby its smallest prime divisor). Among the prime factors (1) some may be equal. , as are natural exponents. We call formula (2) the canonical decomposition of n into prime factors. We have not only proved Theorem 4 but also given a method of obtaining for each natural number n > 1 its canonical decomposition. Theoretically it is always possible to obtain this decomposition for a given natural n > 1, but in practice one may be involved in very difficult and lengthy calculations.

E. the number p) has eleven digits. It is easy to prove that for integral « > 1 π(η—\) «—1 π(η) « where « is a prime number, and π(η— 1) π(ή) «—1 « where « is a composite number. We can prove in an elementary way that the ratio of π(«) to « tends to zero when « tends to infinity. η) = n for any natural n. It is easy to prove that there exist arbitrarily long sequences of natural numbers which contain no prime numbers. , and the last by m+l; thus they are all composite. For m = 100 the numbers would be gigantic, but between the prime numbers 370,261 and 370,373 there lie 111 successive composite numbers.

N2 The columns of this table form an arithmetic progression (with n terms). A. Schinzel advanced the conjecture that if k is a natural number < n not having any common factor > 1 with n, then the &th column of our table will contain at least one prime number. A. Gorzelewski verified this conjecture for all natural numbers n ^ 100. WHAT WE K N O W AND WHAT WE DO NOT K N O W 57 I have put forth the conjecture that every row written in the table (where n > 1) will contain at least one prime number.

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