User:Karsten Meyer/A143548

A143548

Irregular triangle of numbers $\scriptstyle k<p^{2}\,$ such that $\scriptstyle p^{2}\,$ divides $\scriptstyle k^{p-1}-1\,$, with p=prime(n).

p	$p^{2}$	Irregular triangle of numbers k
3	9	1, 8
5	25	1, 7, 18, 24
7	49	1, 18, 19, 30, 31, 48
11	121	1, 3, 9, 27, 40, 81, 94, 112, 118, 120
13	169	1, 19, 22, 23, 70, 80, 89, 99, 146, 147, 150, 168
17	289	1, 38, 40, 65, 75, 110, 131, 134, 155, 158, 179, 214, 224, 249, 251, 288
19	361	1, 28, 54, 62, 68, 69, 99, 116, 127, 234, 245, 262, 292, 293, 299, 307, 333, 360
23	529	1, 28, 42, 63, 118, 130, 170, 177, 195, 255, 263, 266, 274, 334, 352, 359, 399, 411, 466, 487, 501, 528
29	841	1, 14, 41, 60, 63, 137, 190, 196, 221, 236, 267, 270, 374, 416, 425, 467, 571, 574, 605, 620, 645, 651, 704, 778, 781, 800, 827, 840
31	961	1, 115, 117, 145, 229, 235, 333, 338, 374, 388, 414, 430, 439, 440, 448, 513, 521, 522, 531, 547, 573, 587, 623, 628, 726, 732, 816, 844, 846, 960
37	1369	1, 18, 76, 117, 300, 324, 348, 354, 356, 424, 437, 473, 476, 494, 581, 582, 632, 678, 691, 737, 787, 788, 875, 893, 896, 932, 945, 1013, 1015, 1021, 1045, 1069, 1252, 1293, 1351, 1368

A143548, the Euler pseudoprimes and the Fermat pseudoprimes

$\scriptstyle p^{2}\,$ divides $\scriptstyle k^{p-1}-1\,$ could we write as $\scriptstyle k^{p-1}\equiv 1{\pmod {p^{2}}}\,$. The Condition for Euler pseudoprimes is $\scriptstyle k^{\frac {p^{2}-1}{2}}\equiv 1{\pmod {p^{2}}}\,$ in respect of $\scriptstyle p^{2}\,$.