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

ISBN-10: 0080107346

ISBN-13: 9780080107349

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Van der Corput [1]). As regards the conjecture of Goldbach, we observe that it is easy to prove that every natural number > 11 is the sum of two composite numbers. e. a composite number, and n is the sum of two composite numbers 4 and n—A. e. it is composite, and n is the sum of two composite numbers 9 and n—9. We should not conclude from this that the inquiry into composite numbers is easier than investigations about prime numbers. , we have infinitely many composite numbers (so far we only known thirty-eight such composite numbers, of which the greatest is ^1945).

We arrange the natural numbers 1, 2, 3 , . . e. , 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.

2(p—l)=p+(p—2) > p and so the number (p—1) ! + l is > p and by Wilson's theorem is divisible by p; it is therefore a composite number. Hence: if p > 3 is a prime number, then the number (p— 1) ! + l is composite. It follows that there are infinitely many natural numbers n for which n\ + \ is composite. The question then arises whether there also exist infinitely many natural numbers n for which nl + l is prime. We do not know the answer to this problem. The numbers 1 ! + 1 = 2 , 2 ! + l = 3, 3 !

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