## When two 2-digit semiprimes add up to a third semiprime

In this problem, We only consider the squarefree semiprimes (the product of two distinct primes)

When two 2-digit semiprimes add up to a third semiprime.
Each of the six primes is different.

34 + 35 = 69   …..   (2 × 17) + (5 × 7) = 3 × 23
38 + 39 = 77   …..   (2 × 19) + (3 × 13) = 7 × 11
38 + 55 = 93   …..   (2 × 19) + (5 × 11) = 3 × 31
58 + 35 = 93   …..   (2 × 29) + (5 × 7) = 3 × 31
39 + 35 = 74   …..   (3 × 13) + (5 × 7) = 2 × 37

6 = 2 × 3

10 = 2 × 5   ….   14 = 2 × 7   …   15 = 3 × 5   ….   21 = 3 × 7   …..   22 = 2 × 11
26 = 2 × 13   ..   33 = 3 × 11   ..   34 = 2 × 17   ..   35 = 5 × 7   ….   38 = 2 × 19
39 = 3 × 13   ..   46 = 2 × 23   ..   51 = 3 × 17   ..   55 = 5 × 11   ….   57 = 3 × 19
58 = 2 × 29   ..   62 = 2 × 31   ..   65 = 5 × 13   ..   69 = 3 × 23   …   74 = 2 × 37
77 = 7 × 11   ….   82 = 2 × 41   ..   85 = 5 × 17   ..   86 = 2 × 43   …   87 = 3 × 29
91 = 7 × 13   ….   93 = 3 × 31   ..   94 = 2 × 47   ..   95 = 5 × 19

Can you find squarefree semiprimes   A,   B,   C,   D   such that:

A   +   B   =   C
A   –   B   =   D

Each of the prime factors being different?

215 + 6 = 221 = (5 × 43) + (2 × 3) = 13 × 17
215 – 6 = 209 = (5 × 43) – (2 × 3) = 11 × 19

335 + 6 = 341 = (5 × 67) + (2 × 3) = 11 × 31
335 + 6 = 329 = (5 × 67) – (2 × 3) = 7 × 47

371 + 6 = 377 = (7 × 53) + (2 × 3) = 13 × 29
371 – 6 = 365 = (7 × 53) – (2 × 3) = 5 × 73

1357 + 6 = 1363 = (23 × 59) + (2 × 3) = 29 × 47
1357 – 6 = 1351 = (23 × 59) – (2 × 3) = 7 × 193

133 + 10 = 143 = (7 × 19) + (2 × 5) = 11 × 13
133 – 10 = 123 = (7 × 19) – (2 × 5) = 3 × 41

427 + 10 = 437 = (7 × 61) + (2 × 5) = 19 × 23
427 – 10 = 417 = (7 × 61) – (2 × 5) = 3 × 139

1633 + 10 = 1643 = (23 × 71) + (2 × 5) = 31 × 53
1633 – 10 = 1623 = (23 × 71) – (2 × 5) = 3 × 541

129 + 14 = 143 = (3 × 43) + (2 × 7) = 11 × 13
129 – 14 = 115 = (3 × 43) – (2 × 7) = 5 × 23

201 + 14 = 215 = (3 × 67) + (2 × 7) = 5 × 43
201 – 14 = 187 = (3 × 67) – (2 × 7) = 11 × 17

305 + 14 = 319 = (5 × 61) + (2 × 7) = 11 × 29
305 – 14 = 291 = (5 × 61) – (2 × 7) = 3 × 97

183 + 22 = 205 = (3 × 61) + (2 × 11) = 5 × 41
183 – 22 = 161 = (3 × 61) – (2 × 11) = 7 × 23

871 + 22 = 893 = (13 × 67) + (2 × 11) = 19 × 47
871 – 22 = 849 = (13 × 67) – (2 × 11) = 3 × 283

159 + 26 = 185 = (3 × 53) + (2 × 13) = 5 × 37
159 – 26 = 133 = (3 × 53) – (2 × 13) = 7 × 19

2747 + 26 = 2773 = (41 × 67) + (2 × 13) = 47 × 59
2747 – 26 = 2721 = (41 × 67) – (2 × 13) = 3 × 907

111 + 34 = 145 = (3 × 37) + (2 × 17) = 5 × 29
111 – 34 = 77 = (3 × 37) – (2 × 17) = 7 × 11

123 + 38 = 161 = (3 × 41) + (2 × 19) = 7 × 23
123 – 38 = 85 = (3 × 41) – (2 × 19) = 5 × 17

183 + 38 = 221 = (3 × 61) + (2 × 19) = 13 × 17
183 – 38 = 145 = (3 × 61) – (2 × 19) = 5 × 29

215 + 38 = 253 = (5 × 43) + (2 × 19) = 11 × 23
215 – 38 = 177 = (5 × 43) – (2 × 19) = 3 × 59

629 + 38 = 667 = (17 × 37) + (2 × 19) = 23 × 29
629 – 38 = 591 = (17 × 37) – (2 × 19) = 3 × 197

141 + 46 = 187 = (3 × 47) + (2 × 23) = 11 × 17
141 – 46 = 95 = (3 × 47) – (2 × 23) = 5 × 19

201 + 46 = 247 = (3 × 67) + (2 × 23) = 13 × 19
201 – 46 = 155 = (3 × 67) – (2 × 23) = 5 × 31

295 + 46 = 341 = (5 × 59) + (2 × 23) = 11 × 31
295 – 46 = 249 = (5 × 59) – (2 × 23) = 3 × 83

427 + 46 = 473 = (7 × 61) + (2 × 23) = 11 × 43
427 – 46 = 381 = (7 × 61) – (2 × 23) = 3 × 127

583 + 46 = 629 = (11 × 53) + (2 × 23) = 17 × 37
583 – 46 = 537 = (11 × 53) – (2 × 23) = 3 × 179

177 + 58 = 235 = (3 × 59) + (2 × 29) = 5 × 47
177 – 58 = 119 = (3 × 59) – (2 × 29) = 7 × 17

201 + 58 = 259 = (3 × 67) + (2 × 29) = 7 × 37
201 – 58 = 143 = (3 × 67) – (2 × 29) = 11 × 13

469 + 58 = 527 = (7 × 67) + (2 × 29) = 17 × 31
469 – 58 = 411 = (7 × 67) – (2 × 29) = 3 × 137

185 + 62 = 247 = (5 × 37) + (2 × 31) = 13 × 19
185 – 62 = 123 = (5 × 37) – (2 × 31) = 3 × 41

329 + 62 = 391 = (7 × 47) + (2 × 31) = 17 × 23
329 – 62 = 267 = (7 × 47) – (2 × 31) = 3 × 89

129 + 74 = 203 = (3 × 43) + (2 × 37) = 7 × 29
129 – 74 = 55 = (3 × 43) – (2 × 37) = 5 × 11

For more, see Paul’s answer below.

math grad - Interest: Number theory
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### 2 Responses to When two 2-digit semiprimes add up to a third semiprime

1. paul says:

I’ve formatted it a bit better, these are the remainder of the list.

1007 + 74 = 1081 = (19 x 53) + (2 x 37) = 23 x 47
1007 – 74 = 933 = (19 x 53) – (2 x 37) = 3 x 311

177 + 82 = 259 = (3 x 59) + (2 x 41) = 7 x 37
177 – 82 = 95 = (3 x 59) – (2 x 41) = 5 x 19

295 + 82 = 377 = (5 x 59) + (2 x 41) = 13 x 29
295 – 82 = 213 = (5 x 59) – (2 x 41) = 3 x 71

259 + 82 = 341 = (7 x 37) + (2 x 41) = 11 x 31
259 – 82 = 177 = (7 x 37) – (2 x 41) = 3 x 59

201 + 86 = 287 = (3 x 67) + (2 x 43) = 7 x 41
201 – 86 = 115 = (3 x 67) – (2 x 43) = 5 x 23

305 + 86 = 391 = (5 x 61) + (2 x 43) = 17 x 23
305 – 86 = 219 = (5 x 61) – (2 x 43) = 3 x 73

497 + 86 = 583 = (7 x 71) + (2 x 43) = 11 x 53
497 – 86 = 411 = (7 x 71) – (2 x 43) = 3 x 137

407 + 86 = 493 = (11 x 37) + (2 x 43) = 17 x 29
407 – 86 = 321 = (11 x 37) – (2 x 43) = 3 x 107

159 + 94 = 253 = (3 x 53) + (2 x 47) = 11 x 23
159 – 94 = 65 = (3 x 53) – (2 x 47) = 5 x 13

115 + 94 = 209 = (5 x 23) + (2 x 47) = 11 x 19
115 – 94 = 21 = (5 x 23) – (2 x 47) = 3 x 7

205 + 94 = 299 = (5 x 41) + (2 x 47) = 13 x 23
205 – 94 = 111 = (5 x 41) – (2 x 47) = 3 x 37

141 + 106 = 247 = (3 x 47) + (2 x 53) = 13 x 19
141 – 106 = 35 = (3 x 47) – (2 x 53) = 5 x 7

235 + 106 = 341 = (5 x 47) + (2 x 53) = 11 x 31
235 – 106 = 129 = (5 x 47) – (2 x 53) = 3 x 43

217 + 106 = 323 = (7 x 31) + (2 x 53) = 17 x 19
217 – 106 = 111 = (7 x 31) – (2 x 53) = 3 x 37

427 + 106 = 533 = (7 x 61) + (2 x 53) = 13 x 41
427 – 106 = 321 = (7 x 61) – (2 x 53) = 3 x 107

793 + 106 = 899 = (13 x 61) + (2 x 53) = 29 x 31
793 – 106 = 687 = (13 x 61) – (2 x 53) = 3 x 229

1633 + 106 = 1739 = (23 x 71) + (2 x 53) = 37 x 47
1633 – 106 = 1527 = (23 x 71) – (2 x 53) = 3 x 509

183 + 118 = 301 = (3 x 61) + (2 x 59) = 7 x 43
183 – 118 = 65 = (3 x 61) – (2 x 59) = 5 x 13

205 + 118 = 323 = (5 x 41) + (2 x 59) = 17 x 19
205 – 118 = 87 = (5 x 41) – (2 x 59) = 3 x 29

355 + 118 = 473 = (5 x 71) + (2 x 59) = 11 x 43
355 – 118 = 237 = (5 x 71) – (2 x 59) = 3 x 79

259 + 118 = 377 = (7 x 37) + (2 x 59) = 13 x 29
259 – 118 = 141 = (7 x 37) – (2 x 59) = 3 x 47

319 + 118 = 437 = (11 x 29) + (2 x 59) = 19 x 23
319 – 118 = 201 = (11 x 29) – (2 x 59) = 3 x 67

871 + 118 = 989 = (13 x 67) + (2 x 59) = 23 x 43
871 – 118 = 753 = (13 x 67) – (2 x 59) = 3 x 251

177 + 122 = 299 = (3 x 59) + (2 x 61) = 13 x 23
177 – 122 = 55 = (3 x 59) – (2 x 61) = 5 x 11

213 + 122 = 335 = (3 x 71) + (2 x 61) = 5 x 67
213 – 122 = 91 = (3 x 71) – (2 x 61) = 7 x 13

371 + 122 = 493 = (7 x 53) + (2 x 61) = 17 x 29
371 – 122 = 249 = (7 x 53) – (2 x 61) = 3 x 83

185 + 134 = 319 = (5 x 37) + (2 x 67) = 11 x 29
185 – 134 = 51 = (5 x 37) – (2 x 67) = 3 x 17

767 + 134 = 901 = (13 x 59) + (2 x 67) = 17 x 53
767 – 134 = 633 = (13 x 59) – (2 x 67) = 3 x 211

177 + 142 = 319 = (3 x 59) + (2 x 71) = 11 x 29
177 – 142 = 35 = (3 x 59) – (2 x 71) = 5 x 7

235 + 142 = 377 = (5 x 47) + (2 x 71) = 13 x 29
235 – 142 = 93 = (5 x 47) – (2 x 71) = 3 x 31

265 + 142 = 407 = (5 x 53) + (2 x 71) = 11 x 37
265 – 142 = 123 = (5 x 53) – (2 x 71) = 3 x 41

469 + 142 = 611 = (7 x 67) + (2 x 71) = 13 x 47
469 – 142 = 327 = (7 x 67) – (2 x 71) = 3 x 109

Paul.

• benvitalis says:

Thanks. I appreciate that. Lots of typing! I stopped because I was thinking of another puzzle.