20:
642:
28:
467:
458:
386:
828:
274:
143:
85:
689:
198:
732:
323:
637:{\displaystyle v=2{\sqrt {\pi }}{\frac {\Gamma (5/4)}{\Gamma (7/4)}}\prod _{p\equiv 1{\bmod {4}}}{\frac {p-2}{p-1}}\prod _{p\equiv 3{\bmod {4}}}{\frac {p}{p-1}}.}
343:
294:
163:
911:
702:
98:
91:
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
916:
695:
2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601, 2917, 3137, 4357, 5477, 7057, 8101, 8837, 12101, 13457, 14401, 15377, … (sequence
906:
393:
105:
The difficulty in this statement lies in the very sparse nature of this sequence: the number of integers of the form
730:
Friedlander, John; Iwaniec, Henryk (1997), "Using a parity-sensitive sieve to count prime values of a polynomial",
832:
36:
19:
710:
348:
885:
863:
793:
239:
108:
50:
233:
868:
761:
713:) that this set is infinite. However, this is not implied by the Friedlander–Iwaniec theorem.
661:
168:
877:
841:
751:
741:
209:
299:
217:
778:
328:
279:
213:
148:
900:
889:
756:
44:
881:
229:
845:
27:
765:
746:
26:
18:
602:
550:
697:
93:
790:
Heath-Brown, D.R.; Li, Xiannan (2017), "Prime values of
236:
in 2017. In particular, they proved that the polynomial
796:
779:"Iwaniec, Sarnak, and Taylor Receive Ostrowski Prize"
664:
470:
453:{\displaystyle f(n)\sim v{\frac {x^{3/4}}{\log {x}}}}
396:
351:
331:
302:
282:
242:
171:
151:
111:
53:
276:
represents infinitely many primes when the variable
822:
683:
636:
452:
380:
337:
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288:
268:
192:
157:
137:
79:
866:(1998), "Sieving Prime Numbers From Thin Ore",
658:, the Friedlander–Iwaniec primes have the form
8:
220:in part for his contributions to this work.
16:Infinite prime numbers of the form a^2+b^4
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43:states that there are infinitely many
7:
912:Theorems in analytic number theory
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490:
208:The theorem was proved in 1997 by
14:
87:. The first few such primes are
325:is the prime numbers less than
216:. Iwaniec was awarded the 2001
529:
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406:
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312:
306:
1:
917:Theorems about prime numbers
381:{\displaystyle a^{2}+b^{4},}
882:10.1126/science.279.5347.31
823:{\displaystyle a^{2}+p^{4}}
709:It is conjectured (one of
269:{\displaystyle a^{2}+b^{4}}
228:The theorem was refined by
138:{\displaystyle a^{2}+b^{4}}
80:{\displaystyle a^{2}+b^{4}}
41:Friedlander–Iwaniec theorem
933:
846:10.1007/s00222-016-0694-0
833:Inventiones Mathematicae
165:is roughly of the order
684:{\displaystyle a^{2}+1}
193:{\displaystyle X^{3/4}}
907:Additive number theory
824:
747:10.1073/pnas.94.4.1054
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37:analytic number theory
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318:{\displaystyle f(n)}
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864:Cipra, Barry Arthur
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691:, forming the set
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33:
25:
711:Landau's problems
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338:{\displaystyle n}
289:{\displaystyle b}
158:{\displaystyle X}
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740:(4): 1054–1058,
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23:John Friedlander
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214:Henryk Iwaniec
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31:Henryk Iwaniec
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100:
95:
90:
89:
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72:
68:
64:
59:
55:
46:
45:prime numbers
42:
38:
29:
21:
876:(5347): 31,
873:
867:
837:
831:
785:
774:
737:
731:
725:
708:
653:
650:
647:Special case
464:
461:
390:
345:in the form
227:
207:
104:
47:of the form
40:
34:
840:: 441–499,
224:Refinements
901:Categories
717:References
234:Xiannan Li
145:less than
890:118322959
623:−
596:≡
589:∏
579:−
568:−
544:≡
537:∏
513:Γ
491:Γ
483:π
440:
410:∼
766:11038598
869:Science
701:in the
698:A002496
204:History
97:in the
94:A028916
888:
764:
754:
462:where
886:S2CID
757:19742
651:When
388:then
762:PMID
733:PNAS
703:OEIS
232:and
212:and
99:OEIS
39:the
878:doi
874:279
842:doi
838:208
830:",
752:PMC
742:doi
656:= 1
603:mod
551:mod
437:log
35:In
903::
884:,
872:,
836:,
760:,
750:,
738:94
736:,
705:).
200:.
101:).
893:.
880::
849:.
844::
816:4
812:p
808:+
803:2
799:a
769:.
744::
679:1
676:+
671:2
667:a
654:b
632:.
626:1
620:p
616:p
607:4
599:3
593:p
582:1
576:p
571:2
565:p
555:4
547:1
541:p
530:)
527:4
523:/
519:7
516:(
508:)
505:4
501:/
497:5
494:(
478:2
475:=
472:v
444:x
431:4
427:/
423:3
419:x
413:v
407:)
404:n
401:(
398:f
376:,
371:4
367:b
363:+
358:2
354:a
333:n
313:)
310:n
307:(
304:f
284:b
262:4
258:b
254:+
249:2
245:a
186:4
182:/
178:3
174:X
153:X
131:4
127:b
123:+
118:2
114:a
73:4
69:b
65:+
60:2
56:a
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