How many distinct minimal primitive Pythagorean triangles are there with a hypotenuse of 2576450045?

Number Theory Level 5

How many distinct minimal primitive Pythagorean triangles are there with a hypotenuse of 2576450045?

To clarify: The use of "minimal" only refers to sides of the triangle being sorted from smallest to largest. Of course, at least one Pythagorean triangle exists with a hypotenuse of 2576450045. That was checked before posting this problem.


The answer is 64.

A Former Brilliant Member by A Former Brilliant Member

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1 solution

2576450045 = 5 × 13 × 17 × 29 × 37 × 41 × 53 2576450045=5\times 13\times 17\times 29\times 37\times 41 \times 53

The factors are all 4 n + 1 4 n+1 primes. There are 7 distinct such primes. 2 6 = 64 2^6=64 .

See A006278 .

See Representation of integers as sums of two squares

∑ n = 0 m ( m n ) = 2 m \sum _{n=0}^m \binom{m}{n}=2^m

Fermat's theorem on sums of two squares states that "In additive number theory, Fermat's theorem on sums of two squares states that an odd prime p can be expressed as: p = x 2 + y 2 p=x^{2}+y^{2} with x and y integers, if and only if p ≡ 1 ( m o d 4 ) p\equiv 1\pmod {4} ." The definition of a Pythagoroean prime is that it is such a prime. The Brahmagupta–Fibonacci identity "expresses the product of two sums of two squares as a sum of two squares in two different ways. Hence the set of all sums of two squares is closed under multiplication. Specifically, the identity says: ( a 2 + b 2 ) ( c 2 + d 2 ) = ( a d + b c ) 2 + ( a c − b d ) 2 = ( a d − b c ) 2 + ( a c + b d ) 2 \left(a^2+b^2\right) \left(c^2+d^2\right)=(a d+b c)^2+(a c-b d)^2=(a d-b c)^2+(a c+b d)^2 . Since the primes need to used in pairs, one of the primes can not be counted. ∴ \therefore

Here is all 64: 44719563 2576061916 2576450045 70413237 2575487684 2576450045 136188844 2572848117 2576450045 147241676 2572239243 2576450045 170253236 2570818677 2576450045 185727093 2569747124 2576450045 195888364 2568992523 2576450045 249641876 2564327157 2576450045 261457077 2563149436 2576450045 272467083 2562002444 2576450045 310785244 2557637067 2576450045 354364724 2551964043 2576450045 374359797 2549107604 2576450045 476886004 2531930997 2576450045 478355403 2531653796 2576450045 514731147 2524509196 2576450045 531265077 2521081564 2576450045 556386123 2515656836 2576450045 599754357 2505671476 2576450045 637192676 2496413493 2576450045 652093323 2492562764 2576450045 653539844 2492183883 2576450045 678366556 2485540917 2576450045 689329564 2482522827 2576450045 772834084 2457808437 2576450045 809536587 2445965116 2576450045 851138763 2431801316 2576450045 861578997 2428122004 2576450045 897827204 2414953653 2576450045 947278276 2395988043 2576450045 968680636 2387415477 2576450045 978929084 2383231563 2576450045 983035893 2381540524 2576450045 1013117524 2368900107 2576450045 1014492213 2368311716 2576450045 1062959883 2346957844 2576450045 1097970644 2330784267 2576450045 1127400267 2316692356 2576450045 1166751243 2297125676 2576450045 1201004476 2279404107 2576450045 1268493387 2242547516 2576450045 1277352196 2237513397 2576450045 1290799413 2229782884 2576450045 1310700747 2218143004 2576450045 1323927164 2210274123 2576450045 1356862773 2190209636 2576450045 1376312596 2178040053 2576450045 1397969804 2164203147 2576450045 1430107083 2143102556 2576450045 1442983156 2134454133 2576450045 1462025356 2121456267 2576450045 1545325173 2061568564 2576450045 1546414773 2060751364 2576450045 1580263797 2034910604 2576450045 1610545804 2011028853 2576450045 1635884853 1990471196 2576450045 1645040684 1982910987 2576450045 1677589684 1955450763 2576450045 1686203476 1948027893 2576450045 1705550924 1931111307 2576450045 1706671413 1930121116 2576450045 1730978116 1908352587 2576450045 1779167733 1863506644 2576450045 1797667467 1845666956 2576450045 \begin{array}{ccc} 44719563 & 2576061916 & 2576450045 \\ 70413237 & 2575487684 & 2576450045 \\ 136188844 & 2572848117 & 2576450045 \\ 147241676 & 2572239243 & 2576450045 \\ 170253236 & 2570818677 & 2576450045 \\ 185727093 & 2569747124 & 2576450045 \\ 195888364 & 2568992523 & 2576450045 \\ 249641876 & 2564327157 & 2576450045 \\ 261457077 & 2563149436 & 2576450045 \\ 272467083 & 2562002444 & 2576450045 \\ 310785244 & 2557637067 & 2576450045 \\ 354364724 & 2551964043 & 2576450045 \\ 374359797 & 2549107604 & 2576450045 \\ 476886004 & 2531930997 & 2576450045 \\ 478355403 & 2531653796 & 2576450045 \\ 514731147 & 2524509196 & 2576450045 \\ 531265077 & 2521081564 & 2576450045 \\ 556386123 & 2515656836 & 2576450045 \\ 599754357 & 2505671476 & 2576450045 \\ 637192676 & 2496413493 & 2576450045 \\ 652093323 & 2492562764 & 2576450045 \\ 653539844 & 2492183883 & 2576450045 \\ 678366556 & 2485540917 & 2576450045 \\ 689329564 & 2482522827 & 2576450045 \\ 772834084 & 2457808437 & 2576450045 \\ 809536587 & 2445965116 & 2576450045 \\ 851138763 & 2431801316 & 2576450045 \\ 861578997 & 2428122004 & 2576450045 \\ 897827204 & 2414953653 & 2576450045 \\ 947278276 & 2395988043 & 2576450045 \\ 968680636 & 2387415477 & 2576450045 \\ 978929084 & 2383231563 & 2576450045 \\ 983035893 & 2381540524 & 2576450045 \\ 1013117524 & 2368900107 & 2576450045 \\ 1014492213 & 2368311716 & 2576450045 \\ 1062959883 & 2346957844 & 2576450045 \\ 1097970644 & 2330784267 & 2576450045 \\ 1127400267 & 2316692356 & 2576450045 \\ 1166751243 & 2297125676 & 2576450045 \\ 1201004476 & 2279404107 & 2576450045 \\ 1268493387 & 2242547516 & 2576450045 \\ 1277352196 & 2237513397 & 2576450045 \\ 1290799413 & 2229782884 & 2576450045 \\ 1310700747 & 2218143004 & 2576450045 \\ 1323927164 & 2210274123 & 2576450045 \\ 1356862773 & 2190209636 & 2576450045 \\ 1376312596 & 2178040053 & 2576450045 \\ 1397969804 & 2164203147 & 2576450045 \\ 1430107083 & 2143102556 & 2576450045 \\ 1442983156 & 2134454133 & 2576450045 \\ 1462025356 & 2121456267 & 2576450045 \\ 1545325173 & 2061568564 & 2576450045 \\ 1546414773 & 2060751364 & 2576450045 \\ 1580263797 & 2034910604 & 2576450045 \\ 1610545804 & 2011028853 & 2576450045 \\ 1635884853 & 1990471196 & 2576450045 \\ 1645040684 & 1982910987 & 2576450045 \\ 1677589684 & 1955450763 & 2576450045 \\ 1686203476 & 1948027893 & 2576450045 \\ 1705550924 & 1931111307 & 2576450045 \\ 1706671413 & 1930121116 & 2576450045 \\ 1730978116 & 1908352587 & 2576450045 \\ 1779167733 & 1863506644 & 2576450045 \\ 1797667467 & 1845666956 & 2576450045 \\ \end{array}

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This is an interesting problem! As a side note: Given two positive integers x 1 = a 1 2 + b 1 2 x_1=a_1^2+b_1^2 and x 2 = a 2 2 + b 2 2 x_2=a_2^2+b_2^2 , the product of x 1 x 2 x_1x_2 is also a sum of two squares, which is ( a 1 a 2 − b 1 b 2 ) 2 + ( a 1 b 2 − a 2 b 1 ) 2 (a_1a_2-b_1b_2)^2+(a_1b_2-a_2b_1)^2 . Thus, given 2576450045 has 7 prime factors of the form 4 n + 1 4n+1 , we can fix one of the prime as one factor and choose 1, 2, 3, 4, 5 or 6 primes as the other factor, so that we can repeatedly use the fact that the product of two primes of the form 4 n + 1 4n+1 can be represented as a sum of two squares, given that they can be represented as a sum of two squares respectively. This leads to ∑ i = 0 6 ( 6 i ) = 2 6 \sum_{i=0}^6 \binom {6} {i} =2^6 combinations.

ChengYiin Ong - 7 months, 3 weeks ago

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Exactly what I meant by my hand waving explanation.

A Former Brilliant Member - 7 months, 3 weeks ago

I reduced the amount of hand-waving in my explanation.

A Former Brilliant Member - 7 months, 3 weeks ago

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