By Waclaw Sierpinski, I. N. Sneddon, M. Stark
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Additional resources for A Selection of Problems in the Theory of Numbers. Popular Lectures in Mathematics
Van der Corput ). 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 !
A Selection of Problems in the Theory of Numbers. Popular Lectures in Mathematics by Waclaw Sierpinski, I. N. Sneddon, M. Stark