Type: Article
Publication Date: 1971-01-01
Citations: 14
DOI: https://doi.org/10.1090/s0025-5718-1971-0299567-6
Some results in number theory, including the Prime Number Theorem, can be obtained by assuming a random distribution of prime numbers. In addition, conjectural formulae, such as Cherwell’s for the density of prime pairs <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis p comma p plus 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(p,p + 2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> obtained in this way, have been found to agree well with the available evidence. Recently, primes have been determined over ranges of 150,000 numbers with starting points up to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="10 Superscript 15"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>15</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{10^{15}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Statistical arguments are used to obtain a formula for the largest interval between consecutive primes in such a range, and it is found to agree well with recorded values. The same method is applied to predict the maximum interval between consecutive primes occurring below a given integer.