## Squarefree semiprimes (A,B,C); A+B,B+C,C+A

Can you find squarefree semiprimes A, B, C such that each of

A + B
B + C
C + A

is a square number?

Paul found:

$14 + 86 \; = \; (2 \times 7) + (2 \times 43) \; = \; 10^2$
$86 + 35 \; = \; (2 \times 43) + (5 \times 7) \; = 11^2$
$14 + 35 \; = \; (2 \times 7) + (5 \times 7) \; = \; 7^2$

$26 + 74 \; = \; (2 \times 13) + (2 \times 37) \; = \; 10^2$
$74 + 95 \; = \; (2 \times 37) + (5 \times 19) \; = \; 13^2$
$26 + 95 \; = \; (2 \times 13) + (5 \times 19) \; = \; 11^2$

$38 + 106 \; = \; (2 \times 19) + (2 \times 53) \; = \; 12^2$
$106 + 218 \; = \; (2 \times 53) + (2 \times 109) \; = \; 18^2$
$38 + 218 \; = \; (2 \times 19) + (2 \times 109) \; = \; 16^2$

$38 + 158 \; = \; (2 \times 19) + (2 \times 79) \; = \; 14^2$
$158 + 803 \; = \; (2 \times 79) + (11 \times 73) \; = \; 31^2$
$38 + 803 \; = \; (2 \times 19) + (11 \times 73) \; = \; 29^2$

$106 + 218 \; = \; (2 \times 53) + (2 \times 109) \; = \; 18^2$
$218 + 623 \; = \; (2 \times 109) + (7 \times 89) \; = \; 29^2$
$106 + 623 \; = \; (2 \times 53) + (7 \times 89) \; = \; 27^2$

$118 + 206 \; = \; (2 \times 59) + (2 \times 103) \; = \; 18^2$
$206 + 323 \; = \; (2 \times 103) + (17 \times 19) \; = \; 23^2$
$118 + 323 \; = \; (2 \times 59) + (17 \times 19) \; = \; 21^2$

math grad - Interest: Number theory
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### 2 Responses to Squarefree semiprimes (A,B,C); A+B,B+C,C+A

1. paul says:

Here are a few

14 + 86 = 10^2
86 + 35 = 11^2
14 + 35 = 7^2

26 + 74 = 10^2
74 + 95 = 13^2
26 + 95 = 11^2

38 + 106 = 12^2
106 + 218 = 18^2
38 + 218 = 16^2

38 + 158 = 14^2
158 + 803 = 31^2
38 + 803 = 29^2

106 + 218 = 18^2
218 + 623 = 29^2
106 + 623 = 27^2

118 + 206 = 18^2
206 + 323 = 23^2
118 + 323 = 21^2

Paul