<< problem 216 - Investigating the primality of numbers of the ... Skew-cost coding - problem 219 >>

# Problem 218: Perfect right-angled triangles

Consider the right angled triangle with sides a=7, b=24 and c=25. The area of this triangle is 84, which is divisible by the perfect numbers 6 and 28.
Moreover it is a primitive right angled triangle as gcd(a,b)=1 and gcd(b,c)=1.
Also c is a perfect square.

We will call a right angled triangle perfect if

• it is a primitive right angled triangle
• its hypotenuse is a perfect square
We will call a right angled triangle super-perfect if
• it is a perfect right angled triangle and
• its area is a multiple of the perfect numbers 6 and 28.
How many perfect right-angled triangles with c <= 10^16 exist that are not super-perfect?

# My Algorithm

I wrote a small program to check all such triangle with c <= 10^8 and found zero perfect right-angled triangles that are not super-perfect.
See below for the simple code, most of it was copied from problem 86.

Even trying some randomized values for m and n failed to find such a triangle.

Then I went back to my highly valued "paper and pencil" technique and found a relationship:
each primitive triple is defined by
(1) a = m^2 - n^2
(2) b = 2mn
(3) c = m^2 + n^2
(4) gcd(m,n) = 1
(5) (m mod 2) != (n mod 2)
Those equations were already part of multiple Project Euler problems, such as problem 86.

This time there are more restrictions:

• c is a perfect square, so there is an integer d such that c = d^2
• the area of each right-angled triangle is A = ab / 2
• A mod 6 == 0 and A mod 28 == 0
The last restriction can be simplified: the least common multiple lcm(6,28) = 6 * 28 / gcd(6,28) = 84, therefore A mod 84 == 0 and
the area A must be a multiple of 84:
(6) A == 0 mod 84
(7) ab/2 == 0 mod 84
(8) ab == 0 mod 168

I have to find all perfect right-angled triangle where the area is a multiple of 168 (and the hypotenuse < 10^16)

Since c is a perfect square with c = d^2 I can rewrite c = m^2 + n^2 as
(9) d^2 = m^2 + n^2

This is actually a primitive triple again - because of gcd(m, n) = 1. So can repeat the procedure again and there must be some x and y such that
(10) m = x^2 - y^2
(11) n = 2xy
(12) d = x^2 + y^2

Substituting m and n in (1) and (2):
(13) a = (x^2 - y^2)^2 - (2xy)^2
(14) b = 2 * (x^2 - y^2) * 2xy

Therefore the area of these triangles becomes (see (8) ):
(15) ((x^2 - y^2)^2 - (2xy)^2) * 2 * (x^2 - y^2) * 2xy == 0 mod 168
(16) ((x^2 - y^2)^2 - (2xy)^2) * (x^2 - y^2) * xy == 0 mod 42

I wrote two nested loops iterating over all 42^2 basic pairs (x mod 42, y mod 42) (see countNotMod42) - and actually of them produce zero in equation (16).
That means that there are no solutions, no matter whether the limit is 10^8, 10^16 or infinity.

## Alternative Approaches

Nayuki's proof is almost identical to my concept. Some of my ideas are solved by a program (multiple of 42) whereas he showed the same in a mathematical way.
It's also important to note that he found his proof a few years earlier.
I was kind of surprised when my program returned zero and even suspected a bug because I didn't fully trust countNotMod42.

## Note

countNotMod42 fails when x or y are int (or unsigned) instead of long long because of overflows. It cost me half an hour to realize that problem.
This function also contains a constant named Multiplier which can be ignored → set it to 1.

# Interactive test

This feature is not available for the current problem.

# My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too.

       #include <iostream>
#include <cmath>

// return number of pairs (x,y) such that equation (16) is not zero mod 42
unsigned int countNotMod42()
{
unsigned int result = 0;

const auto Multiplier = 1; // no more than 5 because then long long overflows
for (long long x = 0; x < 42 * Multiplier; x++)
for (long long y = 0; y < 42 * Multiplier; y++)
{
auto zero = ((x*x - y*y)*(x*x - y*y) - 2*x*y*2*x*y) * (x*x - y*y) * x * y;
if (zero % 42 != 0)
result++;
}
return result;
}

// greatest common divisor
template <typename T>
T gcd(T x, T y)
{
while (x != 0)
{
auto temp = x;
x = y % x;
y = temp;
}
return y;
}

int main()
{
// count how many possible pairs (x,y) could generate (a,b,c) according to equation (16)
std::cout << countNotMod42() << std::endl;
return 0;

//#define SEARCH
#ifdef  SEARCH
// ---------- try to find a non super-perfect triangle up to 10^8 ---------
// note: code currently not reached because the program exited three lines ago
unsigned long long limit = 100000000;
std::cin >> limit;

unsigned long long countNotSuperPerfect = 0;

// find basic Pythagorean triples (code copied from problem 86)
for (unsigned long long m = 1; m <= sqrt(2*limit); m++)
for (unsigned long long n = 1; n < m; n++)
{
if (m % 2 == n % 2)
continue;
if (gcd(m, n) != 1)
continue;

// two sides
auto a = m*m - n*n;
auto b = 2*m*n;
auto c = m*m + n*n;
if (c > limit)
break;

// is c a perfect square ?
unsigned long long cRoot = sqrt(c);
if (cRoot * cRoot != c)
continue;

// is area a multiple of 6 and 28 ?
auto area = a * b / 2;
if (area % 6 != 0 || area % 28 != 0) // can be combined: area % lcm(6, 28) = area % 84
{
countNotSuperPerfect++;
continue;
}
}

// show result
std::cout << countNotSuperPerfect << std::endl;
return 0;
#endif
}


This solution contains 11 empty lines, 11 comments and 4 preprocessor commands.

# Benchmark

The correct solution to the original Project Euler problem was found in less than 0.01 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=gnu++11 -DORIGINAL)

See here for a comparison of all solutions.

Note: interactive tests run on a weaker (=slower) computer. Some interactive tests are compiled without -DORIGINAL.

# Changelog

August 28, 2017 submitted solution

# Difficulty

Project Euler ranks this problem at 55% (out of 100%).

# Heatmap

Please click on a problem's number to open my solution to that problem:

 green solutions solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too yellow solutions score less than 100% at Hackerrank (but still solve the original problem easily) gray problems are already solved but I haven't published my solution yet blue solutions are relevant for Project Euler only: there wasn't a Hackerrank version of it (at the time I solved it) or it differed too much orange problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte red problems are not solved yet but I wrote a simulation to approximate the result or verified at least the given example - usually I sketched a few ideas, too black problems are solved but access to the solution is blocked for a few days until the next problem is published [new] the flashing problem is the one I solved most recently

I stopped working on Project Euler problems around the time they released 617.
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200
 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300
 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400
 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500
 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600
 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637
The 310 solved problems (that's level 12) had an average difficulty of 32.6% at Project Euler and
I scored 13526 points (out of 15700 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.

My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.

Look at my progress and performance pages to get more details.

 << problem 216 - Investigating the primality of numbers of the ... Skew-cost coding - problem 219 >>
more about me can be found on my homepage, especially in my coding blog.
some names mentioned on this site may be trademarks of their respective owners.
thanks to the KaTeX team for their great typesetting library !