Friedlander–Iwaniec theorem

The Friedlander–Iwaniec theorem proves that infinitely many primes have the form a² + b⁴, where a and b are positive integers.

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Sieve theory: A collection of methods for estimating or counting integers that avoid specified prime divisors. The theorem adapts sieve methods to the sparse values of a² + b⁴.
Prime number: An integer greater than one whose only positive divisors are one and itself. The theorem asserts primality for infinitely many values of its polynomial expression.
John Friedlander: A Canadian number theorist known for work in analytic number theory and sieve methods. He co-authored the theorem establishing infinitely many primes of the stated form.
Bunyakovsky conjecture: A conjecture predicting infinitely many prime values for irreducible integer polynomials satisfying necessary local conditions. The theorem proves infinitude for a notable two-variable polynomial, echoing a broader prime-values problem.
Parity problem: A limitation of classical sieve methods: they struggle to distinguish numbers with an even or odd number of prime factors. Overcoming this sieve limitation is central to proving the values include primes.
Polynomial: An expression built from variables and coefficients using addition, subtraction, and multiplication. The primes in the theorem arise as values of a specific two-variable polynomial.
Henryk Iwaniec: A Polish-American mathematician whose work spans analytic number theory and sieve methods. He co-authored the proof with John Friedlander.
Bateman–Horn conjecture: A conjecture predicting asymptotic counts for simultaneous prime values of polynomials. It suggests quantitative estimates far beyond the theorem's infinitude conclusion.
Exponential sum: A finite sum of complex exponentials whose phases encode arithmetic information. Exponential-sum estimates supply cancellation needed beyond basic sieve bounds.
Sum of two squares: An integer represented as x² + y² for integers x and y. The theorem's representation resembles this classical problem but uses a fourth power.