MEI Maths Item of the Month January 2015
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MEI Maths Item of the Month January 2015 Happy 2015: A Triple of Triples 2015 is the product of 3 distinct primes: 5×13×31 2014 and 2013 are also the product 3 distinct primes. Can you find a smaller triple (n, n+1, n+2) where n, n+1 and n+2 are all the product of 3 distinct primes? Are there any quadruples (n, n+1, n+2, n+3) where n, n+1, n+2 and n+3 are all the product of 3 distinct primes? Solution It is useful to answer the second question first. Every set of 4 consecutive numbers contains a multiple of 4. A multiple of 4 will have 2 at least twice in its prime factorisation, therefore there cannot be any quadruples that are all the product of 3 distinct primes. This is useful for searching for triples that are the product of 3 distinct primes as any cases (n, n+1, n+2) as it means that the middle of the triple must be an even number of the form 2p1p2 where p1 and p2 are distinct primes. A table of all such numbers up to p2 = 41: 3 3 5 7 11 13 17 19 23 29 31 37 5 30 7 42 70 11 66 110 154 13 78 130 182 286 17 102 170 238 374 442 19 114 190 266 418 494 646 23 29 31 37 41 138 174 186 222 246 230 290 310 370 410 322 406 434 518 574 506 638 682 814 902 598 754 806 962 1066 782 986 1054 1258 1394 874 1102 1178 1406 1558 1334 1426 1702 1886 1798 2146 2378 2294 2542 3034 This is the table for values of n+1. The tables for n and n+2 can then be inspected to see which values are the products of 3 primes. 1 of 3 TB v1.0 © MEI 27/03/2015 MEI Maths Item of the Month The table for n = 2p1p2 – 1 3 3 5 7 11 13 17 19 23 29 31 37 5 29 7 41 69 11 65 109 153 13 77 129 181 285 17 101 169 237 373 441 19 113 189 265 417 493 645 11 67 13 79 131 17 19 171 239 191 267 419 495 647 23 29 31 37 41 137 173 185 221 245 229 289 309 369 409 321 405 433 517 573 505 637 681 813 901 597 753 805 961 1065 781 985 1053 1257 1393 873 1101 1177 1405 1557 1333 1425 1701 1885 1797 2145 2377 2293 2541 3033 The table for n = 2p1p2 + 1 3 3 5 7 11 13 17 19 23 29 31 37 5 7 71 155 287 443 23 323 507 599 783 875 29 31 187 311 291 407 639 683 755 807 987 1055 1103 1179 1335 1427 1799 37 223 371 519 815 963 1259 1407 1703 2147 2295 41 247 575 903 1067 1395 1559 1887 2379 2543 3035 Prime numbers are shaded red and values > 2015 are shaded purple. 2 of 3 TB v1.0 © MEI 27/03/2015 MEI Maths Item of the Month Checking the triples not yet discounted: 153 285 169 265 493 321 505 781 873 289 405 637 753 985 1333 185 805 1053 1177 1797 369 517 813 961 1257 1701 245 573 901 1065 1393 1885 154 286 170 266 494 322 506 782 874 290 406 638 754 986 1334 186 806 1054 1178 1798 370 518 814 962 1258 1702 246 574 902 1066 1394 1886 155 287 171 267 495 323 507 783 875 291 407 639 755 987 1335 187 807 1055 1179 1799 371 519 815 963 1259 1703 247 575 903 1067 1395 1887 153 is a multiple of 9 287=7×41 169=13^2 265=5×53 493=17×29 321=3×107 505=5×101 781=11×71 873 is a multiple of 9 289=17² 405 is a multiple of 9 637=7²×13 753=3×251 985=5×197 1333=31×43 185=5×37 807=3×269 1053 is a multiple of 9 1177=11×107 1797=3×599 369 is a multiple of 9 517=11×47 813=3×271 961=21² 1257=3×419 1701 is a multiple of 9 245=5×7² 573=3×191 901=17×53 1067=11×97 1393=7×199 All three are the product of 3 primes 1885, 1886 and 1887 are all the product of 3 primes. A smaller solution than this can be found: 1309, 1310, 1311. 3 of 3 TB v1.0 © MEI 27/03/2015
