## Integers n for which n^2 + 1 is a product of 3 different primes

Find positive integers   $n$   for which   $n^2 + 1$   is a …

(1)   …   product of three different primes:

n = 13, 17, 21, 23, and 27

$13^2 \; + \; 1 \; = \; 170 \; = \; 2 \; \times \; 5 \; \times \; 17$
$17^2 \; + \; 1 \; = \; 290 \; = \; 2 \; \times \; 5 \; \times \; 29$
$21^2 \; + \; 1 \; = \; 442 \; = \; 2 \; \times \; 13 \; \times \; 17$
$23^2 \; + \; 1 \; = \; 530 \; = \; 2 \; \times \; 5 \; \times \; 53$
$27^2 \; + \; 1 \; = \; 730 \; = \; 2 \; \times \; 5 \; \times \; 73$

(2)   …   product of three different odd primes

n = 112

$112^2 \; + \; 1 \; = \; 12545 \; = \; 5 \; \times \; 13 \; \times \; 193$

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math grad - Interest: Number theory
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### One Response to Integers n for which n^2 + 1 is a product of 3 different primes

1. paul says:

Here are a few more

13^2 + 1 = 170 = 2 x 5 x 17
17^2 + 1 = 290 = 2 x 5 x 29
21^2 + 1 = 442 = 2 x 13 x 17
23^2 + 1 = 530 = 2 x 5 x 53
27^2 + 1 = 730 = 2 x 5 x 73
31^2 + 1 = 962 = 2 x 13 x 37
33^2 + 1 = 1090 = 2 x 5 x 109
37^2 + 1 = 1370 = 2 x 5 x 137
43^2 + 1 = 1850 = 2 x 5 x 37
53^2 + 1 = 2810 = 2 x 5 x 281
55^2 + 1 = 3026 = 2 x 17 x 89
57^2 + 1 = 3250 = 2 x 5 x 13
63^2 + 1 = 3970 = 2 x 5 x 397
67^2 + 1 = 4490 = 2 x 5 x 449
72^2 + 1 = 5185 = 5 x 17 x 61
75^2 + 1 = 5626 = 2 x 29 x 97
77^2 + 1 = 5930 = 2 x 5 x 593
81^2 + 1 = 6562 = 2 x 17 x 193
87^2 + 1 = 7570 = 2 x 5 x 757
89^2 + 1 = 7922 = 2 x 17 x 233
91^2 + 1 = 8282 = 2 x 41 x 101
93^2 + 1 = 8650 = 2 x 5 x 173
97^2 + 1 = 9410 = 2 x 5 x 941
98^2 + 1 = 9605 = 5 x 17 x 113
99^2 + 1 = 9802 = 2 x 13 x 29

And part 2.

72^2 + 1 = 5185 = 5 x 17 x 61
98^2 + 1 = 9605 = 5 x 17 x 113
112^2 + 1 = 12545 = 5 x 13 x 193
122^2 + 1 = 14885 = 5 x 13 x 229
128^2 + 1 = 16385 = 5 x 29 x 113
132^2 + 1 = 17425 = 5 x 17 x 41
138^2 + 1 = 19045 = 5 x 13 x 293
142^2 + 1 = 20165 = 5 x 37 x 109
148^2 + 1 = 21905 = 5 x 13 x 337
162^2 + 1 = 26245 = 5 x 29 x 181
172^2 + 1 = 29585 = 5 x 61 x 97
174^2 + 1 = 30277 = 13 x 17 x 137
192^2 + 1 = 36865 = 5 x 73 x 101
200^2 + 1 = 40001 = 13 x 17 x 181

Paul.