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

 
 

Find more examples.

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

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.

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