Three topics that are not related mathematically, “An Infinite Class of Rogers-Ramanujan Type Continued Fractions,” “Representation of Integers by Sum of Squares of Primes,” and “Numerical Evidence Relating to the Riemann and Mertens Hypotheses,” are all based on discoveries made in Cardiff using FORTRAN programs on the Honeywell system. One or more theorems have emerged in each case, and in one area some conjectures emerged as well. The author did not attempt to find machine proofs of any of the theorems, but did employ the machine in interactive mode to seek additional evidence. This approach was of considerable help in finding all the proofs.
The Rogers-Ramanujan continued fraction formula for | x | < 1 is given by The author developed some computer programs to search for continued fractions such as (1) where the product on the left-hand side is of the form where a , b , c , d , and M are positive integers and ( a + b ) = ( c + d ) = M ≤ 24
The programs produced three possible identities, such as (1), which the author already knew, as well as another two identities, which the author did not know about. The author later proved these identities using Watson’s method [1].
Recently, Churchhouse modified the original programs to extend the search for continued fractions to cover cases where the product on the left contained an arbitrary number of factors in the numerator and denominator, rather than two. The programs, therefore, examine products such as
The programs are interactive; after the values of M, ai, and b i had been read in, the first n terms of the Taylor series obtained from (3) were presented on the screen.
For over 200 years it has been known that every positive integer is the sum of four squares [2]. For even longer it has been conjectured that every even integer is the sum of two primes and that every odd integer is the sum of three primes [2]. The principal new conjecture is that every positive integer is representable as the sum of the squares of eight primes. The program that led to this conjecture also provided an unexpected theorem. The program calculated how many integers in a given range are not representable as the sum of the squares of k primes for k = 2, 3, 4,…. When the program reached k = 8, it reported that there are no integers from 1 to 20,000 that are not so representable. This evidence formed the basis for the conjecture. Recently, the author has used to computer to study the rate of growth in the autoclave series.
The last topic is the Riemann and Mertens hypotheses and is related to the previous work of Good and Churchhouse [3]. The Mertens hypothesis is that M ( N ) < &sqrt;N for N > 2 where and &mgr; ( n ) is the Moebius function, μ ( n ) = ( - 1 ) k if n is the product of k distinct primes, &mgr; ( 1 ) = 1, and &mgr;( n )=0 otherwise. Mertens’s hypothesis is related to the Riemann hypothesis in that, if M ( N ) = O ( N .5 + &egr; ) then the Riemann hypothesis is true [4]. Counts of M ( N ) led the author to conclude that &mgr; ( n ) can be considered as a pseudo-random variable taking the values: Several examples led the author to examine the values of M k where for k = O ( 1 ) 49,000. The pseudo-random variable model predicted that 7688 values of M k would lie in the range of –1 to –10, and the author’s examination found 7513 such values (see Good and Churchhouse [3] for a detailed table). If n were randomly distributed with the above probabilities, the Central Limit Theorem could be applied and the important conclusion would be which implies that the Mertens hypothesis is false.
