WOLFRAM|DEMONSTRATIONS PROJECT

Pseudoprime

​
base a
77
base b
50
pseudoprime
base 77
4
703
2465
5073
7633
13051
38
741
2701
5461
7709
16017
39
969
2806
5713
7957
16745
57
1105
2965
5785
10081
17081
65
1387
3705
6305
10585
20501
76
1513
4033
6441
10777
21953
247
1653
4371
6533
11229
22971
285
1891
4636
6541
12871
23591
pseudoprime
base 77 and 50
2701
5461
29341
30889
46657
66397
91001
pseudoprime
base 50
21
357
931
2107
3913
8041
49
561
1037
2499
3977
8113
51
637
1281
2501
4753
8911
119
697
1649
2701
5461
9061
133
793
1729
2821
5719
9073
147
817
2009
2989
6601
9331
231
833
2041
3201
7693
9457
301
861
2047
3281
7701
9577
Fermat's little theorem (FLT) states that for any prime number
p
and coprime base
a
,
p
a
≡a(modp)
. If this congruence fails, then
p
cannot be prime. Using FLT as a primality test seems promising because it distinguishes primes from nonprimes in many cases. For example,
3
2
≡8≡2(mod3)
,
5
2
≡32≡2(mod5)
,
7
2
≡128≡2(mod7)
,
9
2
≡512≡8(mod9)
, so 9 isn't prime, and so on. This primality test works up to
341
2
≡2(mod341)
, but
341=11×31
. A pseudoprime like 341 is a composite number that passes a primality test. Different bases lead to different pseudoprimes.