This paper describes how the even Goldbach conjecture was confirmed to be true for all even numbers not larger than 4 • 10 18. Using a result of Ramaré and Saouter, it follows that the odd Goldbach conjecture is true up to 8.37 • 10 26. 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.
For q an odd prime power with q > 169 we prove that there are always three consecutive primitive elements in the finite field F q . Indeed, there are precisely eleven values of q ≤ 169 for which this is false. For 4 ≤ n ≤ 8 we present conjectures on the size of q 0 (n) such that q > q 0 (n) guarantees the existence of n consecutive primitive elements in F q , provided that F q has characteristic at least n. Finally, we improve the upper bound on q 0 (n) for all n ≥ 3.
We prove that for all q > 61, every non-zero element in the finite field F q can be written as a linear combination of two primitive roots of F q . This resolves a conjecture posed by Cohen and Mullen.
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