Friedlander–Iwaniec theorem

Infinite prime numbers of the form a^2+b^4 From Wikipedia, the free encyclopedia

Friedlander–Iwaniec theorem

In analytic number theory the Friedlander–Iwaniec theorem states that there are infinitely many prime numbers of the form . The first few such primes are

2, 5, 17, 37, 41, 97, 101, 137, 181, 197, 241, 257, 277, 281, 337, 401, 457, 577, 617, 641, 661, 677, 757, 769, 821, 857, 881, 977, … (sequence A028916 in the OEIS).
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John Friedlander
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Henryk Iwaniec

The difficulty in this statement lies in the very sparse nature of this sequence: the number of integers of the form less than is roughly of the order .

History

The theorem was proved in 1997 by John Friedlander and Henryk Iwaniec.[1] Iwaniec was awarded the 2001 Ostrowski Prize in part for his contributions to this work.[2]

Refinements

Summarize
Perspective

The theorem was refined by D.R. Heath-Brown and Xiannan Li in 2017.[3] In particular, they proved that the polynomial represents infinitely many primes when the variable is also required to be prime. Namely, if is the prime numbers less than in the form then

where

Special case

When b = 1, the Friedlander–Iwaniec primes have the form , forming the set

2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601, 2917, 3137, 4357, 5477, 7057, 8101, 8837, 12101, 13457, 14401, 15377, … (sequence A002496 in the OEIS).

It is conjectured (one of Landau's problems) that this set is infinite. However, this is not implied by the Friedlander–Iwaniec theorem.

References

Further reading

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