Final Report Summary - PRIMEGAPS (Gaps between primes and almost primes. Patterns in primes and almost primes. Approximations to the twin prime and Goldbach conjectures.)
The main goals of the project were
(i) to achieve a good approximation of the famous, 2300 years old twin prime problem, which asserts that two neighboring odd numbers n and n+2 can be simultaneously primes for infinitely many values of n and
(ii) to connect these results with the famous Green-Tao theorem according to which there are arbitrarily long (finite) arithmetic progressions in the sequence of primes.
Thus, one of the main problems of the project was to show the existence of a number d with the property that we should have arbitrarily long arithmetic progressions of primes so that p+d would be also prime for every element p of the progression. However, this assertion immediately implies the existence of infinitely many bounded gaps between primes, the so called Bounded Gap Conjecture. This was only showed by D. Goldston, the PI and C. Yildirim under an unproved conjecture, according to which primes of size X are statistically uniformly distributed in arithmetic progressions of difference less than X^c with an exponent c>1/2. Hence the original goal was to show under the same condition the existence of arbitrarily long arithmetic progressions of generalized twin prime pairs {p, p+d} for some d. In May 2013 Y. Zhang showed the existence of infinitely many bounded gaps between consecutive primes (the bound was 70,000,000.) The PI succeeded in generalizing Zhang’s theorem and this more general result made possible to show the above fact unconditionally, even in the stronger form that p and p+d are consecutive primes for all elements of the progression (after the original conditional result was proved by the PI in 2010.)
De Polignac conjectured in 1849 that for every even h, n and n+h can be consecutive primes infinitely often. We will call such numbers h Polignac numbers. However until this year it was not known whether any Polignac number exists at all. The proof of the Bounded Gap Conjecture showed the existence of at least one Polignac number h. The PI succeeded to show by the above generalization of Zhang’s result that there are infinitely many Polignac numbers, there lower density is positive and the differences between consecutive Polignac numbers are uniformly bounded.
In joint work with Goldston and Yildirim we showed that for an arbitrarily small c>0 the prime gaps less than clogp form a positive proportion of all gaps. In a joint work with B. Farkas and Sz. Gy. Révész (another team member) we found the optimal weight function in our method (the so called GPY sieve) and thereby we both improved and found the limit of the original GPY method producing gaps of size less than (logp)^c for any c>3/7. When after the result of Zhang, T. Tao launched a Polymath Project to improve Zhang’s bound, the above result of us, as well as various additional ideas of the PI and G. Harcos (another team member) contributed to the success of the project, to push down the bound below5,000. A manuscript of more than 160 pages is in the final stage of preparation at present. In collaboration with D. Goldston, S.W.Graham and C. Yildirim we succeeded to find a unified method which showed in a stronger and more general form some famous conjectures of Erdős and Erdős-Mirsky, about consecutive integers for which the divisor function and other important arithmetic functions take the same value.
The above mentioned generalizations of the result of Zhang by the PI made possible for him to show a 65 years old conjecture of Erdős according to which the ratio of two consecutive prime-gaps can take arbitrarily large and arbitrarily small positive values as well.
(i) to achieve a good approximation of the famous, 2300 years old twin prime problem, which asserts that two neighboring odd numbers n and n+2 can be simultaneously primes for infinitely many values of n and
(ii) to connect these results with the famous Green-Tao theorem according to which there are arbitrarily long (finite) arithmetic progressions in the sequence of primes.
Thus, one of the main problems of the project was to show the existence of a number d with the property that we should have arbitrarily long arithmetic progressions of primes so that p+d would be also prime for every element p of the progression. However, this assertion immediately implies the existence of infinitely many bounded gaps between primes, the so called Bounded Gap Conjecture. This was only showed by D. Goldston, the PI and C. Yildirim under an unproved conjecture, according to which primes of size X are statistically uniformly distributed in arithmetic progressions of difference less than X^c with an exponent c>1/2. Hence the original goal was to show under the same condition the existence of arbitrarily long arithmetic progressions of generalized twin prime pairs {p, p+d} for some d. In May 2013 Y. Zhang showed the existence of infinitely many bounded gaps between consecutive primes (the bound was 70,000,000.) The PI succeeded in generalizing Zhang’s theorem and this more general result made possible to show the above fact unconditionally, even in the stronger form that p and p+d are consecutive primes for all elements of the progression (after the original conditional result was proved by the PI in 2010.)
De Polignac conjectured in 1849 that for every even h, n and n+h can be consecutive primes infinitely often. We will call such numbers h Polignac numbers. However until this year it was not known whether any Polignac number exists at all. The proof of the Bounded Gap Conjecture showed the existence of at least one Polignac number h. The PI succeeded to show by the above generalization of Zhang’s result that there are infinitely many Polignac numbers, there lower density is positive and the differences between consecutive Polignac numbers are uniformly bounded.
In joint work with Goldston and Yildirim we showed that for an arbitrarily small c>0 the prime gaps less than clogp form a positive proportion of all gaps. In a joint work with B. Farkas and Sz. Gy. Révész (another team member) we found the optimal weight function in our method (the so called GPY sieve) and thereby we both improved and found the limit of the original GPY method producing gaps of size less than (logp)^c for any c>3/7. When after the result of Zhang, T. Tao launched a Polymath Project to improve Zhang’s bound, the above result of us, as well as various additional ideas of the PI and G. Harcos (another team member) contributed to the success of the project, to push down the bound below5,000. A manuscript of more than 160 pages is in the final stage of preparation at present. In collaboration with D. Goldston, S.W.Graham and C. Yildirim we succeeded to find a unified method which showed in a stronger and more general form some famous conjectures of Erdős and Erdős-Mirsky, about consecutive integers for which the divisor function and other important arithmetic functions take the same value.
The above mentioned generalizations of the result of Zhang by the PI made possible for him to show a 65 years old conjecture of Erdős according to which the ratio of two consecutive prime-gaps can take arbitrarily large and arbitrarily small positive values as well.