The polynomial $X^2+Y^4$ captures its primes

John Friedlander and Henryk Iwaniec

Review from Zentralblatt MATH:

The question of showing that a specified polynomial over the integers represents infinitely many primes is a very old one, but until now the only successes had been in instances where $N(x)$, the number of values of the polynomial not exceeding a large number $x$, is very nearly as large as $x$. Besides the very classical cases of linear polynomials $an+b$ and binary quadratic forms $am^2+bmn+cn^2$ one might mention the general quadratic polynomial in two variables discussed by {\it H. Iwaniec} [Acta Arith. 24, 435-459 (1974; Zbl 0287.12019)]. Some of the other ideas used are drawn from earlier papers of the authors. The reader is not required to be familiar with automorphic forms, but it is suggested that it was essential that the authors were, as otherwise the paper could not have been written in its present form.

One of the by-products of the authors' work is given by them as a separate theorem, as follows. They show $$ \mathop{\sum\sum}_{r^2+s^2=p\le x}\biggl( {s \over r} \biggr) \ll x^{76/77} ,$$ in which the summand is a Jacobi symbol in which $r$ is odd. Thus there is a degree of equidistribution among the ``spins'' of primes $p$ given by the Jacobi symbols under consideration.

Reviewed by G.Greaves

Keywords: prime representing polynomials; parity; bilinear form hypothesis; asymptotic formula; asymptotic sieve form primes; harmonic analysis; lattice point problem; quartic region; grössencharacters; Siegel's theorem; equidistribution; Jacobi symbols

Classification (MSC2000): 11N35 11N32 11N36 11P21

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