Primes in intervals of bounded length

<p>The infamous <italic>twin prime conjecture</italic> states that there are infinitely many pairs of distinct primes which differ by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Until recently this conjecture had seemed to be far out of reach with current techniques. However, in April 2013, Yitang Zhang proved the existence of a finite bound <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that there are infinitely many pairs of distinct primes which differ by no more than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This is a massive breakthrough, making the twin prime conjecture look highly plausible, and the techniques developed help us to better understand other delicate questions about prime numbers that had previously seemed intractable.</p> <p>Zhang even showed that one can take <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B equals 70000000"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo>=</mml:mo> <mml:mn>70000000</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">B = 70000000</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Moreover, a co-operative team, <italic>Polymath8</italic>, collaborating only online, had been able to lower the value of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4680"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>4680</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">{4680}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. They had not only been more careful in several difficult arguments in Zhang’s original paper, they had also developed Zhang’s techniques to be both more powerful and to allow a much simpler proof (and this forms the basis for the proof presented herein).</p> <p>In November 2013, inspired by Zhang’s extraordinary breakthrough, James Maynard dramatically slashed this bound to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="600"> <mml:semantics> <mml:mn>600</mml:mn> <mml:annotation encoding="application/x-tex">600</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, by a substantially easier method. Both Maynard and Terry Tao, who had independently developed the same idea, were able to extend their proofs to show that for any given integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m greater-than-or-equal-to 1"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">m\geq 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there exists a bound <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Subscript m"> <mml:semantics> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">B_m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that there are infinitely many intervals of length <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Subscript m"> <mml:semantics> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">B_m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> containing at least <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> distinct primes. We will also prove this much stronger result he ...