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.

Answer 3

In your examples, $GCD(20572,2859471,2859545)=37$ but the other is primitive and included as the last member of this list of primitives. You can find the other hundreds of multiples of primitives by plugging in all the factors of $2859545$ into the formula and then multiplying each term by the respective cofactors.

For all valid $C$-values, there are $2^{n-1}$ matching primitive Pythagorean triples where $n$ is the number of prime factors of $C$. There may be multiples of triple too. $$2859545=5×13×29×37×41\quad\text{ (5 distinct prime factors)}$$

Here is how you find at least the primitive triples, using Euclid's formula. $$f(m,k)=\quad A=m^2-k^2\qquad B=2mk\qquad C=m^2+k^2$$

Matching side C using $F(m,k)$ \begin{equation} C=m^2+k^2\implies k=\sqrt{C-m^2}\\ \text{for}\qquad \bigg\lfloor\frac{ 1+\sqrt{2C-1}}{2}\bigg\rfloor \le m \le \lfloor\sqrt{C-1}\rfloor \end{equation} The lower limit ensures $m>k$ and the upper limit ensures $k\in\mathbb{N}$. $$C=65\implies \bigg\lfloor\frac{ 1+\sqrt{130-1}}{2}\bigg\rfloor=6 \le m \le \lfloor\sqrt{65-1}\rfloor=8\\ \land \quad m\in\{7,8\}\Rightarrow k\in\{4,1\}\\$$ $$F(7,4)=(33,56,65)\qquad \qquad F(8,1)=(63,16,65) $$

Below are $2^4=16$ triples found by this method.

$$f(1219,1172)=(112377,2857336,2859545)$$ $$f(1292,1091)=(478983,2819144,2859545)$$ $$f(1348,1021)=(774663,2752616,2859545)$$ $$f(1403,944)=(1077273,2648864,2859545)$$ $$f(1411,932)=(1122297,2630104,2859545)$$ $$f(1436,893)=(1264647,2564696,2859545)$$ $$f(1504,773)=(1664487,2325184,2859545)$$ $$f(1564,643)=(2032647,2011304,2859545)$$ $$f(1576,613)=(2108007,1932176,2859545)$$ $$f(1597,556)=(2241273,1775864,2859545)$$ $$f(1637,424)=(2499993,1388176,2859545)$$ $$f(1648,379)=(2572263,1249184,2859545)$$ $$f(1667,284)=(2698233,946856,2859545)$$ $$f(1669,272)=(2711577,907936,2859545)$$ $$f(1688,101)=(2839143,340976,2859545)$$ $$f(1691,8)=(2859417,27056,2859545)$$