How many Pythagorean triangles which have hypotenuse equal to $2859545$

Question

By using the trail and error, I could find these triangle $$20572, 2859471, 2859545$$$$27056, 2859417, 2859545$$ I couldn't continue to find the others triangles because they need more time. Is there an easy method to find the others triangles?

Answer 1

$c = 2859545$ factors as $5 \times 13 \times 29 \times 37 \times 41$. Each of these primes is congruent to $1$ mod $4$, so they factor over the Gaussian integers: $$5 = \left( 1+2\,i \right) \left( 1-2\,i \right) , 13 = \left( 3+2\,i \right) \left( 3-2\,i \right) , 29 = \left( 5+2\,i \right) \left( 5-2\, i \right) , \\37 = \left( 1+6\,i \right) \left( 1-6\,i \right) , 41 = \left( 5+4 \,i \right) \left( 5-4\,i \right)$$ For each of the five primes $p = (a_p + i b_p)(a_p - i b_p)$, let $f_p$ be either $(a_p + i b_p)^2$, $(a_p + i b_p)(a_p - i b_p)$, or $(a_p - i b_p)^2$. Then $f_1 f_2 \ldots f_5$ is a Gaussian integer $x + i y$ such that $x^2 + y^2 = c^2$. There are $3^5 = 243$ possibilities, but we won't count the trivial case $x = c$, $y=0$, and taking account of complex conjugation that leaves $242/2 = 121$ different solutions.

$$ \begin {array}{cccc} \left\{ 20572,2859471 \right\} & \left\{ 24388,2859441 \right\} & \left\{ 27056,2859417 \right\} & \left\{ 54636,2859023 \right\} \\ \left\{ 78329, 2858472 \right\} & \left\{ 102705,2857700 \right\} & \left\{ 112377, 2857336 \right\} & \left\{ 151536,2855527 \right\} \\ \left\{ 157287,2855216 \right\} & \left\{ 159951 ,2855068 \right\} & \left\{ 163761,2854852 \right\} & \left\{ 184295, 2853600 \right\} \\ \left\{ 233044,2850033 \right\} & \left\{ 241900,2849295 \right\} & \left\{ 262392,2847481 \right\} & \left\{ 290567,2844744 \right\} \\ \left\{ 296296,2844153 \right\} & \left\{ 311025,2842580 \right\} & \left\{ 335257,2839824 \right\} & \left\{ 340976,2839143 \right\} \\ \left\{ 389455,2832900 \right\} & \left\{ 398257 ,2831676 \right\} & \left\{ 412920,2829575 \right\} & \left\{ 446600, 2824455 \right\} \\ \left\{ 466908,2821169 \right\} & \left\{ 473304,2820103 \right\} & \left\{ 478983,2819144 \right\} & \left\{ 493580,2816625 \right\} \\ \left\{ 517584,2812313 \right\} & \left\{ 527100,2810545 \right\} & \left\{ 550375,2806080 \right\} & \left\{ 574287,2801284 \right\} \\ \left\{ 594425,2797080 \right\} & \left\{ 603889 ,2795052 \right\} & \left\{ 627705,2789800 \right\} & \left\{ 642148, 2786511 \right\} \\ \left\{ 647759,2785212 \right\} & \left\{ 654073,2783736 \right\} & \left\{ 703888,2771559 \right\} & \left\{ 727500,2765455 \right\} \\ \left\{ 730080,2764775 \right\} & \left\{ 774663,2752616 \right\} & \left\{ 777231,2751892 \right\} & \left\{ 782772,2750321 \right\} \\ \left\{ 806200,2743545 \right\} & \left\{ 820401 ,2739332 \right\} & \left\{ 852977,2729364 \right\} & \left\{ 861455, 2726700 \right\} \\ \left\{ 875568,2722201 \right\} & \left\{ 881049,2720432 \right\} & \left\{ 907936,2711577 \right\} & \left\{ 927420,2704975 \right\} \\ \left\{ 946856,2698233 \right\} & \left\{ 950456,2696967 \right\} & \left\{ 952972,2696079 \right\} & \left\{ 1001167,2678556 \right\} \\ \left\{ 1023975,2669920 \right\} & \left\{ 1049191,2660112 \right\} & \left\{ 1074801,2649868 \right\} & \left\{ 1077273,2648864 \right\} \\ \left\{ 1080807,2647424 \right\} & \left\{ 1099825,2639580 \right\} & \left\{ 1122297,2630104 \right\} & \left\{ 1152920,2616825 \right\} \\ \left\{ 1171716,2608463 \right\} & \left\{ 1175196,2606897 \right\} & \left\{ 1193920,2598375 \right\} & \left\{ 1202708,2594319 \right\} \\ \left\{ 1238159,2577588 \right\} & \left\{ 1243348,2575089 \right\} & \left\{ 1249184,2572263 \right\} & \left\{ 1264647,2564696 \right\} \\ \left\{ 1295111,2549448 \right\} & \left\{ 1308300,2542705 \right\} & \left\{ 1319175,2537080 \right\} & \left\{ 1337393,2527524 \right\} \\ \left\{ 1362348,2514161 \right\} & \left\{ 1367409,2511412 \right\} & \left\{ 1380400,2504295 \right\} & \left\{ 1388176,2499993 \right\} \\ \left\{ 1401708,2492431 \right\} & \left\{ 1430705,2475900 \right\} & \left\{ 1451769,2463608 \right\} & \left\{ 1456728,2460679 \right\} \\ \left\{ 1469455,2453100 \right\} & \left\{ 1498575,2435420 \right\} & \left\{ 1516057,2424576 \right\} & \left\{ 1521551,2421132 \right\} \\ \left\{ 1564724,2393457 \right\} & \left\{ 1585080,2380025 \right\} & \left\{ 1587300,2378545 \right\} & \left\{ 1607528,2364921 \right\} \\ \left\{ 1624500,2353295 \right\} & \left\{ 1632456,2347783 \right\} & \left\{ 1652420,2333775 \right\} & \left\{ 1664487,2325184 \right\} \\ \left\{ 1671705,2320000 \right\} & \left\{ 1699225,2299920 \right\} & \left\{ 1711116,2291087 \right\} & \left\{ 1715727,2287636 \right\} \\ \left\{ 1735175,2272920 \right\} & \left\{ 1737295,2271300 \right\} & \left\{ 1773772,2242929 \right\} & \left\{ 1775864,2241273 \right\} \\ \left\{ 1796784,2224537 \right\} & \left\{ 1834545,2193500 \right\} & \left\{ 1843920,2185625 \right\} & \left\{ 1855217,2176044 \right\} \\ \left\{ 1859596,2172303 \right\} & \left\{ 1881009,2153788 \right\} & \left\{ 1896455,2140200 \right\} & \left\{ 1903097,2134296 \right\} \\ \left\{ 1914639,2123948 \right\} & \left\{ 1932176,2108007 \right\} & \left\{ 1939300,2101455 \right\} & \left\{ 1954368,2087449 \right\} \\ \left\{ 1957152,2084839 \right\} & \left\{ 1972100,2070705 \right\} & \left\{ 1991604,2051953 \right\} & \left\{ 2011304,2032647 \right\} \\ \left\{ 2015908,2028081 \right\} &&&\\ \end{array} $$

Answer 2

Factor $2859545 = 5 \cdot 13 \cdot 29 \cdot 37 \cdot 41$.

All these primes are of the form $4k+1$ and so can be expressed as sums of two squares in essentially one way.

You can combine the solutions for each prime into several different solutions for $2859545$ using Brahmagupta's identity.