913:. Moreover, they proved that this bound is also sharp. In other words, coefficients larger than 49598666989151226098104244512918 do not guarantee irreducibility. The method of Filaseta and Gross was also generalized to provide similar sharp bounds for some other bases by Cole, Dunn, and Filaseta.
593:
398:
235:
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274:
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71:
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118:
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966:
476:
281:
127:
1197:
Cole, Morgan; Dunn, Scott; Filaseta, Michael (2016). "Further irreducibility criteria for polynomials with non-negative coefficients".
775:
A further generalization of the theorem allowing coefficients larger than digits was given by
Filaseta and Gross. In particular, let
1148:
748:
1001:
769:
1255:
764:. It is clear from context that the "A. Cohn" mentioned by Polya and Szegő is Arthur Cohn (1894–1940), a student of
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75:
38:
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17:
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42:
739:
1227:
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761:
470:
103:
1244:
921:
98:
765:
82:
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1182:
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993:
34:
1141:
Mathematicians
Fleeing from Nazi Germany: Individual Fates and Global Impact
1056:
1113:
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948:
form the representation of a prime number in that base. This is the
588:{\displaystyle p(x)=a_{k}x^{k}+a_{k-1}x^{k-1}+\cdots +a_{1}x+a_{0}}
393:{\displaystyle f(x)=a_{m}x^{m}+a_{m-1}x^{m-1}+\cdots +a_{1}x+a_{0}}
230:{\displaystyle p=a_{m}10^{m}+a_{m-1}10^{m-1}+\cdots +a_{1}10+a_{0}}
940:
is an irreducible polynomial with integer coefficients that have
804:
be a polynomial with non-negative integer coefficients such that
74:—that is, for it to be unfactorable into the product of lower-
1143:. Princeton, N.J.: Princeton University Press. p. 346.
738:
The base 10 version of the theorem is attributed to Cohn by
944:
1, then there exists a base such that the coefficients of
439:
The theorem can be generalized to other bases as follows:
1128:
Arthur Cohn's entry at the
Mathematics Genealogy Project
27:
Sufficient condition for a polynomial to be unfactorable
888:
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106:
48:
952:and its truth or falsity remains an open question.
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661:
632:
587:
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268:
229:
112:
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1049:Aufgaben und Lehrsätze aus der Analysis, Bd 2
8:
1164:Filaseta, Michael; Gross, Samuel S. (2014).
1065:Problems and theorems in analysis, volume 2
994:"Prime Numbers and Irreducible Polynomials"
916:An analogue of the theorem also holds for
93:The criterion is often stated as follows:
1181:
1112:
1097:"On an irreducibility theorem of A. Cohn"
1014:
890:
889:
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853:49598666989151226098104244512918, then
18:A. Cohn's irreducibility criterion
1067:. Vol. 2. Springer. p. 137.
752:while the generalization to any base
7:
1228:"A. Cohn's irreducibility criterion"
1139:Siegmund-Schultze, Reinhard (2009).
1063:Pólya, George; Szegő, Gábor (2004).
1047:Pólya, George; Szegő, Gábor (1925).
768:who was awarded his doctorate from
633:{\displaystyle 0\leq a_{i}\leq b-1}
1166:"49598666989151226098104244512918"
833:is prime. If all coefficients are
25:
967:Perron's irreducibility criterion
749:Problems and Theorems in Analysis
269:{\displaystyle 0\leq a_{i}\leq 9}
33:is a sufficient condition for a
1101:Canadian Journal of Mathematics
31:Cohn's irreducibility criterion
936:of this criterion is that, if
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1002:American Mathematical Monthly
906:{\displaystyle \mathbb {Z} }
770:Frederick William University
722:{\displaystyle \mathbb {Z} }
428:{\displaystyle \mathbb {Z} }
66:{\displaystyle \mathbb {Z} }
1272:
595:is a polynomial such that
1183:10.1016/j.jnt.2013.11.001
918:algebraic function fields
1170:Journal of Number Theory
1061:English translation in:
942:greatest common divisor
669:is a prime number then
462:{\displaystyle b\geq 2}
1114:10.4153/CJM-1981-080-0
962:Eisenstein's criterion
950:Bunyakovsky conjecture
907:
876:
847:
827:
798:
734:History and extensions
723:
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634:
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276:) then the polynomial
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231:
114:
67:
1211:10.4064/aa8376-5-2016
1091:; Filaseta, Michael;
908:
877:
848:
846:{\displaystyle \leq }
828:
826:{\displaystyle f(10)}
799:
756:is due to Brillhart,
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1051:. Springer, Berlin.
886:
882:is irreducible over
875:{\displaystyle f(x)}
857:
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797:{\displaystyle f(x)}
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691:{\displaystyle p(x)}
673:
662:{\displaystyle p(b)}
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128:
104:
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1256:Theorems in algebra
992:Murty, Ram (2002).
903:
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719:
698:is irreducible in
688:
659:
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585:
459:
425:
404:is irreducible in
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266:
227:
110:
63:
1074:978-3-540-63686-1
113:{\displaystyle p}
78:polynomials with
16:(Redirected from
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1199:Acta Arithmetica
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1089:Brillhart, John
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1016:10.1.1.225.8606
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922:finite fields
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747:
737:
443:Assume that
438:
99:prime number
92:
83:coefficients
30:
29:
1251:Polynomials
1205:: 137–181.
766:Issai Schur
39:irreducible
1245:Categories
1233:PlanetMath
1037:(dvi file)
973:References
35:polynomial
1176:: 16–49.
1011:CiteSeerX
841:≤
772:in 1921.
625:−
619:≤
606:≤
554:⋯
543:−
527:−
454:≥
359:⋯
348:−
332:−
261:≤
248:≤
196:⋯
185:−
169:−
89:Statement
1095:(1981).
1057:73165700
956:See also
934:converse
928:Converse
758:Filaseta
1033:2695645
762:Odlyzko
237:(where
122:base 10
80:integer
1147:
1071:
1055:
1031:
1013:
760:, and
76:degree
37:to be
1029:JSTOR
997:(PDF)
920:over
744:Szegő
740:Pólya
640:. If
469:is a
97:If a
1145:ISBN
1069:ISBN
1053:OCLC
932:The
742:and
473:and
1207:doi
1203:175
1178:doi
1174:137
1109:doi
1021:doi
1007:109
746:in
124:as
41:in
1247::
1230:.
1201:.
1172:.
1168:.
1105:33
1103:.
1099:.
1027:.
1019:.
1005:.
999:.
980:^
924:.
818:10
212:10
178:10
149:10
85:.
1236:.
1213:.
1209::
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1180::
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1111::
1077:.
1059:.
1035:.
1023::
946:p
938:p
901:]
898:x
895:[
891:Z
870:)
867:x
864:(
861:f
821:)
815:(
812:f
792:)
789:x
786:(
783:f
754:b
729:.
717:]
714:x
711:[
707:Z
686:)
683:x
680:(
677:p
657:)
654:b
651:(
648:p
628:1
622:b
614:i
610:a
603:0
581:0
577:a
573:+
570:x
565:1
561:a
557:+
551:+
546:1
540:k
536:x
530:1
524:k
520:a
516:+
511:k
507:x
501:k
497:a
493:=
490:)
487:x
484:(
481:p
457:2
451:b
435:.
423:]
420:x
417:[
413:Z
386:0
382:a
378:+
375:x
370:1
366:a
362:+
356:+
351:1
345:m
341:x
335:1
329:m
325:a
321:+
316:m
312:x
306:m
302:a
298:=
295:)
292:x
289:(
286:f
264:9
256:i
252:a
245:0
223:0
219:a
215:+
207:1
203:a
199:+
193:+
188:1
182:m
172:1
166:m
162:a
158:+
153:m
143:m
139:a
135:=
132:p
108:p
61:]
58:x
55:[
51:Z
20:)
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