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Number Theory Seminar

Parametrization and Spacing of Roots of Cubic Congruences

Matthew Welsh (Rutgers)

Location:  Hill 525
Date & time: Tuesday, 24 April 2018 at 2:00PM - 3:00PM

Roots of quadratic congruences, for example x^2 = -1 (mod m), can be parametrized using binary quadratic forms. This parametrization leads to an approximation, to error less than 1/m, of x/m by a fraction with denominator of size m^{1/2}. This in turn shows that those roots with m at most M are spaced by at least 1/M, a fact crucial to Fouvry and Iwaniec's proof that there are infinitely many primes of the form x^2 + p^2.

In this talk we will discuss similar parametrization, approximation, and spacing results for roots of cubic congruences, focusing on x^3 = 2 (mod m). A main result is that inside any square with side length 1/M in R^2/Z^2, there are O(1) points (x/m, x^2/m) mod 1 with x^3 = 2 (mod m) and m ~M.

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