Robert E. Dressler

Sums of distinct primes from congruence classes modulo 12

It is shown that every integer greater than 1969, 1349, 1387, 1475 is a sum of distinct primes of the form 12k + 1, 12k + 5, 12k + 7, 12k + 11, respectively. Furthermore, these lower bounds are best possible. In [2], A. M~kowski proved the following: THEOREM. Every integer greater than 55, 121, 161, 205 is a sum of distinct primes of the form 4k 1, 4k + 1, 6k 1, 6k + 1, respectively. Furthermore, these lower bounds are best possible. To do this, he used the work of Breusch [1] and the following theorem of H. E. Richert [4]. THEOREM. Let ml, M2, ... be an infinite increasing sequence of positive integers such that, for some positive integer k, the inequality mi+I k. Suppose there exists a nonnegative integer a, such that the numbers a + 1, a + 2, * *, a + mk+1 are all expressible as a sum of distinct members of the set {ml, m2, , ink k} Then, every integer greater than a is expressible as a sum of distinct members of the sequence min, M2,* Our purpose is to strengthen M~kowskis work by using Richerts theorem and the following theorem of Molsen [3]. THEOREM. For n _ 118, the interval (n, 4n/3) contains a prime of each of the forms 12k + 1, 12k + 5, 12k + 7, 12k + 11. Our work was done on the IBM 360/50 computer at Kansas State University and we wish to thank Mr. Gary Schmidt for his help in the programming. The program was written in FORTRAN and the computer time was 12.42 minutes. So much time was used because we wanted more information than we actually needed to prove the theorem. For example, we obtained information about the multiplicities of representation. We now give our result. THEOREM. Every integer greater than 1969, 1349, 1387, 1475 is a sum of distinct primes of the form 12k + 1, 12k + 5, 12k + 7, 12k + 11, respectively. Furthermore, these lower bounds are best possible. Proof. In the case of 12k + 1, put a 1969, mk = 2029, and mk+l = 2053. In the case of 12k + 5, put a = 1349, mk= 1301, and m,+1 = 1361. In the case of 12k + 7, put a = 1387, m, = 1327, and m,+1 = 1399. In the case of 12k + 11, put a = 1475, mk = 1523, andmk+l = 1559. Received June 20, 1973. AMS (MOS) subject classifications (1970). Primary 10J15.

Gaps between integers with the same prime factors

We give numerical and theoretical evidence in support of the conjecture of Dressler that between any two positive integers having the same prime factors there is a prime. In particular, it is shown that the abc conjecture implies that the gap between two consecutive such numbers a > a 1/2- ∈ , and it is shown that this lower bound is best possible. Dresslers conjecture is verified for values of a and c up to 7. 10 13 .

An elementary proof of a theorem of Erdös on the sum of divisors function

Abstract Let σ(n) be the sum of the positive divisors of the positive integer n. We give an elementary proof of the following theorem due to P. Erdos: If, g(x), is the number of positive integers, m, such that, σ(m) ≤ x, then there is a positive constant, csuch that, g(x) = cx + o(x). In addition we derive c= Π p {(1− 1 p )(1 +( 1 p + 1) + ( 1 p 2 + p + 1) + ( 1 p 3 + 1 p 2 + p + 1) + …)} .

Some remarks concerning the number theoretic functions () and Ω()

In this paper we establish a duality relation between the number theoretic functions w(n) and fl(/i), and we investigate some of its consequences, one of which concerns the Riemann zeta function.