# a-file for OEIS sequence A184763, "Irregular triangle in which row # n has the values of k>n such that sum_{i=n..k} i^2 is a square." # # For a given pair of values n and k, let m be the integer whose # square is the sum referred to in the above definition. We then # have # # n^2 + (n+1)^2 + (n+2)^2 + ... + k^2 = m^2 (Eqn. 1) # # This can be expressed as # # m^2 = (1^2 + 2^2 + ... + k^2) - (1^2 + 2^2 + ... + (n-1)^2) # # and since 1^2 + 2^2 + ... + k^2 = (2*k^3 + 3*k^2 + k)/6, Eqn. 1 can # be rewritten as # # m^2 = (2*k^3 + 3*k^2 + k)/6 - s (Eqn. 2) # # where # # s = (2*(n-1)^3 + 3*(n-1)^2 + (n-1))/6 # # Multiplying each side of Eqn. 2 by 72^2 = 5184 gives # # 5184*m^2 = 1728*k^3 + 2592*k^2 + 864*k - 5184*s # or # (72*m)^2 = (12*k)^3 + 18*(12*k)^2 + 72*(12*k) - 5184*s # # and substituting x=12*k and y=72*m gives the elliptic curve # in a standard form as # # y^2 = x^3 + 18*x^2 + 72*x - 5184*s # # for which the integer solutions can be readily found using Magma; # e.g., copying the Magma code # # for n in [1..100] do # s:=(2*(n-1)^3 + 3*(n-1)^2 + (n-1)) div 6; # A:=IntegralPoints(EllipticCurve([0,18,0,72,-5184*s])); # for i in [1..#A] do # x:=Round(Abs(A[i][1])); # y:=Round(Abs(A[i][2])); # if (x mod 12 eq 0) and (y mod 72 eq 0) then # k:=x div 12; # m:=y div 72; # if (k gt n) then # n, k, m; # end if; # end if; # end for; # end for; # # and pasting it into the Online Magma Calculator at # # http://magma.maths.usyd.edu.au/calc/ # # will cause it to -- # # -- find, for each value of n in 1..100, all the integer solutions # of the above elliptic curve (i.e., the points (x,y) where both # x and y are integers), and # # -- for each of those (x,y) pairs in which x/12 and y/72 (i.e., # k and m) are both integers and k > n, output on a single line # the values of n, k, and m. # # The output below was obtained by using the Online Magma Calculator # (which, at present, has a 120-second time limit per run) to search # through all values of n in the interval 1..460 (using multiple # runs, breaking the interval into a number of smaller subintervals # by changing the limits in the "for" statement at the top of the # above code). It contains every ordered triple of integers (n,k,m) # that satisfies Eqn. 1 (with k > n) for n in the interval 1..460. # 1 24 70 3 4 5 3 580 8075 3 963 17267 7 29 92 7 39 143 7 56 245 7 190 1518 7 2215 60207 9 32 106 9 552057 236818619 11 22908 2001863 13 108 652 15 111 679 15 326 3406 17 39 138 17 5345 225643 18 28 77 20 21 29 20 43 158 20 308 3128 20 1221044 778998480 21 116 724 22 80 413 22 6910 331668 25 48 182 25 50 195 25 73 357 25 578 8033 25 624 9010 25 3625 126035 25 21624 1835940 27 59 253 27 364 4017 28 77 385 28 123 788 30 198 1612 32 609 8687 32 4087 150878 32 61281 8758575 32 148856 33158210 38 48 143 38 96 531 38 349 3770 38 686 10384 38 11918 751228 44 67 274 44 93 495 50 171 1281 50 15674 1133000 52 147 1012 52 389 4433 55 3533 121268 58 2132 56855 60 92 440 60 3238 106403 64 305 3069 65 282 2725 65 928 16332 67 116 655 67 8516 453765 73 194 1525 73 22873 1997277 74 36554 4035066 76 99 430 83 276 2619 83 26003 2420957 87 136 795 91 332 3465 103 142231 30969419 104 967 17364 104 10185 593489 106 402 4620 106 1379 29575 112 305 3007 112 35080 3793482 115 7265 357550 117 789 12787 117 1797 43993 119 120 169 121 144 650 124 173 1055 128 3306 109769 129 4522 175591 131 852 14345 132 430 5083 132 41340 4852918 137 232 1828 140 428 5032 142 3287 108823 146 3578 123588 158 4551 177281 168 217 1365 168 466 5681 168 1856 46166 168 2559 74750 168 52608 6966638 168 803210146 13142656332949 170 466 5676 172 710 10857 175 512 6565 175 5503 235717 175 2065611775 54201639547685 178 361 3726 181 213 1133 183 589 8140 183 1240 25185 183 3438 116402 183 57303 7919743 192 279 2222 192 2842 87483 193 288 2372 197 220 1022 197 949 16817 197 2204 59738 197 2965 93223 197 3459 117468 197 119547 23864378 199 487 6001 200 17888 1381338 204 925 16169 210 585 7990 210 4603 180323 214 752 11781 216 311 2596 216 553 7293 216 2549341296 74315832006486 223 1470 32500 225 298 2257 225 312 2530 225 631 8954 225 70449 10795835 227 259 1397 232 1289 26657 233 10866 653991 234 1194 23746 244 364 3366 246 1519 34125 247 36310 3994732 248 343 2908 253 806 13019 253 79213 12871767 255 783 12443 255 976 17461 255 3279 108405 258 906 15576 262 1967 50327 269 1885 47201 272 8552 456638 280 631 8756 286 13510 906660 287 336 2205 287 575 7463 287 2047 53417 291 8243 432113 294 367 2849 294 6149 278404 301 326 1599 301 925 15975 301 1574 35945 302 855 14127 302 94550 16785532 304 327 1546 304 655 9196 306 12604 817005 309 837 13639 311 10392 611665 312 14736 1032830 312 24527 2217782 316 2164 58050 319 61568 8820175 319 6048560 8588501901 321 1856 46064 322 1019 18497 322 100810 18479832 323 2531 73461 324 44725 5460999 331 18043 1399327 344 1065 19741 346 5922 263120 349 5677 246959 353 376 1786 358 718 10412 377 1074 19893 377 118025 23410093 378 6355 292495 379 428 2855 379 1522 34034 379 31578 3239860 380 9979 575560 390 45126 5534606 390 96101 17200244 395 972 16915 395 2603 76563 395 16236 1194469 403 29554 2933420 409 226224 62122486 413 1299 26610 413 129293 26841353 429 121412 24424994 433 721 9911 433 5089 209565 433 6216 282934 433 23568 2088988 442 514 4088 443 1010 17750 451 1508 33373 454 479 2379 454 1350 28106 456 466 1529 456 1033 18343