abc conjecture

Abstract: This paper describes how the even Goldbach conjecture was confirmed to be true for all even numbers not larger than 4 · 1018. Using a result of Ramaré and Saouter, it follows that the odd Goldbach conjecture is true up to 8.37 · 1026. The empirical data collected during this extensive verification effort, namely, counts and first occurrences of so-called minimal Goldbach partitions with a given smallest prime and of gaps between consecutive primes with a given even gap, are used to test several conjectured formulas related to prime numbers. In particular, the counts of minimal Goldbach partitions and of prime gaps are in excellent accord with the predictions made using the prime k-tuple conjecture of Hardy and Littlewood (with an error that appears to be O( √ t log log t), where t is the true value of the quantity being estimated). Prime gap moments also show excellent agreement with a generalization of a conjecture made in 1982 by Heath-Brown. The Goldbach conjecture [13] is a famous mathematical problem whose proof, or disproof, has so far resisted the passage of time [20, Problem C1]. (According to [1], Waring and, possibly, Descartes also formulated similar conjectures.) It states, in its modern even form, that every even number larger than four is the sum of two odd prime numbers, i.e., that n = p + q. Here, and in what follows, n will always be an even integer larger than four, and p and q will always be odd prime numbers. The additive decomposition n = p + q is called a Goldbach partition of n. The one with the smallest p will be called the minimal Goldbach partition of n; the corresponding p will be denoted by p(n) and the corresponding q by q(n). It is known that up to a given number x at most O(x) even integers do not have a Goldbach partition [30], and that every large enough even number is the sum of a prime and the product of at most two primes [24]. Furthermore, according to [48], every odd number greater that one is the sum of at most five primes. As described in Table 1, over a time span of more than a century the even Goldbach conjecture was confirmed to be true up to ever-increasing upper limits. Section 1 describes the methods that were used by the first author, with computational help from the second and third authors, and others, to set the limit of verification of the Goldbach conjecture at 4 · 10. Section 2 presents a small subset of the empirical data that was gathered during the verification, namely, counts and first occurrences of primes in minimal Goldbach partitions, and counts and first occurrences of prime gaps, and compares it with the predictions made by Received by the editor May 21, 2012 and, in revised form, December 6, 2012. 2010 Mathematics Subject Classification. Primary 11A41, 11P32, 11N35; Secondary 11N05, 11Y55.

Abstract: Given f in Z[x_1,...,x_n], we compute the density of x in Z^n such that f(x) is squarefree, assuming the abc conjecture. Given f,g in Z[x_1,...,x_n], we compute unconditionally the density of x in Z^n such that gcd(f(x),g(x))=1. Function field analogues of both results are proved unconditionally. Finally, assuming the abc conjecture, given f in Z[x], we estimate the size of the image of f({1,2,...,n}) in (Q^*/Q^*2) union {0}.