## Number Theory Seminar

- Event Type
- Seminar/Symposium
- Sponsor
- Department of Mathematics
- Location
- Altgeld Hall 347
- Date
- Mar 5, 2024 11:00 - 11:50 am
- Speaker
- Mikhail Gabdullin (UIUC)
- Contact
- Jesse Thorner and Yuan Liu
- jat9@illinois.edu and yyyliu@illinois.edu
- Views
- 29
- Originating Calendar
- Number Theory Seminar Calendar
Title: Long strings of consecutive composite values of polynomials

Abstract: There is a famous and still open conjecture of Bunyakovsky that for any irreducible polynomial f over the integers (with a positive leading coefficient and no common factor for its values) there are infinitely many n for which f(n) is prime. Since we do not know how to attack it, one can try to think in the opposite direction: how many consecutive numbers n, n+1,...,n+m up to some threshold X we are able to construct so that all the values f(n), f(n+1),...,f(n+m) are composite? Some simple argument implies that this is possible for m of order log X. In a recent work of Ford, Konyagin, Tao, Maynard and Pomerance it was shown that one can take m to be as large as (log X)(log log X)^{C(f)}, where the exponent C(f) is exponentially small in the degree of f. We make this bound independent on f and improve it to (log X)(log log X)^{1/835}. This is a joint work with Kevin Ford.