Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4⋅10¹⁸

Abstract

This paper describes how the even Goldbach conjecture was confirmed to be true for all even numbers not larger than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4 dot 10 Superscript 18"> <mml:semantics> <mml:mrow> <mml:mn>4</mml:mn> <mml:mo> ⋅ </mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>18</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">4\cdot 10^{18}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Using a result of Ramaré and Saouter, it follows that the odd Goldbach conjecture is true up to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="8.37 dot 10 Superscript 26"> <mml:semantics> <mml:mrow> <mml:mn>8.37</mml:mn> <mml:mo> ⋅ </mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>26</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">8.37\cdot 10^{26}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -tuple conjecture of Hardy and Littlewood (with an error that appears to be <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis StartRoot t log log t EndRoot right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msqrt> <mml:mi>t</mml:mi> <mml:mi>log</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mi>log</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mi>t</mml:mi> </mml:msqrt> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O(\sqrt {t\log \log t})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t"> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding="application/x-tex">t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the true value of the quantity being estimated). Prime gap moments also show excellent agreement with a generalization of a conjecture made in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1982"> <mml:semantics> <mml:mn>1982</mml:mn> <mml:annotation encoding="application/x-tex">1982</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by Heath-Brown.

The authors' abstract, as published at the source. Mathematics of Computation, 2013 · DOI ↗