All four primitive Pythagorean quadruples with single-digit values
A Pythagorean quadruple is a tuple of integers such that:
![{\ displaystyle a, b, c, d \ in \ mathbb {Z}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/850ea0fcdd7e86e90df9b636cff1829b6e787b7e)
-
.
These are the solutions to a Diophantine equation . Most of the time, only positive whole numbers are considered as solutions.
Primitive Pythagorean Quadruples
A Pythagorean quadruple is called a primitive Pythagorean quadruple if the values are positive integers and the greatest common divisor of the four values is 1 (if so ). Each Pythagorean quadruple is an integral multiple of a primitive Pythagorean quadruple.
![{\ displaystyle (a, b, c, d)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1aa9ecfbf0677a3d15f751b1f7dfae0adb24512)
![{\ displaystyle \ operatorname {ggT} (a, b, c, d) = 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cadc1a9f56381aee6f085d431f99f48f882b7ae)
Example 1:
- The tuple is a primitive Pythagorean quadruple because is and holds.
![{\ displaystyle (a, b, c, d) = (2,3,6,7)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6edea7872c28b5836db2853303d6ef9ca76d76b4)
![{\ displaystyle \ operatorname {gcd} (2,3,6,7) = 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3cb0bc07db06a76864ad5fef816c78d1bcc62dac)
![{\ displaystyle 2 ^ {2} + 3 ^ {2} + 6 ^ {2} = 7 ^ {2} \ quad (= 49)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbf71e8dba0032c7ddad8dc4e4f0d0b432b1606d)
Example 2:
- The tuple is not a primitive Pythagorean quadruple because is, although holds.
![{\ displaystyle (a, b, c, d) = (5 \ cdot 2.5 \ cdot 3.5 \ cdot 6.5 \ cdot 7) = (10.15,30,35)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13f4a81a787ef69ffe108396e559ae3e9989fcbb)
![{\ displaystyle \ operatorname {gcd} (10,15,30,35) = 5 \ not = 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ab92a8f91f358322cb92b3e52c0da84cddaf2c7)
![{\ displaystyle 10 ^ {2} + 15 ^ {2} + 30 ^ {2} = 35 ^ {2} \ quad (= 1225)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/738f36e5cb60900d2c8c6dc135a744193dcf4815)
Examples
There are 31 primitive Pythagorean quadruples, all of which have values less than 30:
|
|
|
|
|
1 |
2 |
2 |
3 |
|
2 |
3 |
6th |
7th |
|
1 |
4th |
8th |
9 |
|
4th |
4th |
7th |
9 |
|
2 |
6th |
9 |
11 |
|
6th |
6th |
7th |
11 |
|
3 |
4th |
12 |
13 |
|
2 |
5 |
14th |
15th |
|
|
|
|
|
|
|
2 |
10 |
11 |
15th |
|
1 |
12 |
12 |
17th |
|
8th |
9 |
12 |
17th |
|
1 |
6th |
18th |
19th |
|
6th |
6th |
17th |
19th |
|
6th |
10 |
15th |
19th |
|
4th |
5 |
20th |
21st |
|
4th |
8th |
19th |
21st |
|
|
|
|
|
|
|
4th |
13 |
16 |
21st |
|
8th |
11 |
16 |
21st |
|
3 |
6th |
22nd |
23 |
|
3 |
14th |
18th |
23 |
|
6th |
13 |
18th |
23 |
|
9 |
12 |
20th |
25th |
|
12 |
15th |
16 |
25th |
|
2 |
7th |
26th |
27 |
|
|
|
|
|
|
|
2 |
10 |
25th |
27 |
|
2 |
14th |
23 |
27 |
|
7th |
14th |
22nd |
27 |
|
10 |
10 |
23 |
27 |
|
3 |
16 |
24 |
29 |
|
11 |
12 |
24 |
29 |
|
12 |
16 |
21st |
29 |
|
|
Any number of other non-primitive Pythagorean quadruples can be formed from these primitive Pythagorean quadruples. For example, one can from the primitive Pythagorean quadruple by multiplication by the non-primitive Pythagorean quadruple , , etc. form.
![{\ displaystyle (a, b, c, d) = (1,2,2,3)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7189a36316c414cab4124f9c7d580c44af80a5a5)
![{\ displaystyle 2,3,4, \ ldots}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89baa5838d3c4659b9f9fa14f2ead70ac6246990)
![{\ displaystyle (a, b, c, d) = (2,4,4,6)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cb62490f3647b3d8ae8e8d2f3721f0617a6b7e2)
![{\ displaystyle (a, b, c, d) = (3,6,6,9)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb8e2d8589f0fd78c6548de3a0f1e4c49f498953)
![{\ displaystyle (a, b, c, d) = (4,8,8,12)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc18502a763b256ff6bfbbdc7a81a855f4f717c3)
Geometric interpretation
A Pythagorean quadruple defines a parallelepiped with integral side lengths and (with the amount of being meant). The space diagonal of this cuboid then has an integral length . Pythagorean quadruples are therefore also called Pythagorean boxes in English .
![{\ displaystyle (a, b, c, d)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1aa9ecfbf0677a3d15f751b1f7dfae0adb24512)
![{\ displaystyle | a |, | b |}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86e7f6f137f37986cdce1beba48879b24dde9057)
![{\ displaystyle | c |}](https://wikimedia.org/api/rest_v1/media/math/render/svg/befca0a8a04f484ff29d1b75e530e622b4d7bd28)
![| a |](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b61d5baa05004815f3abc52f517ce62b609b9b6)
![a](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)
![{\ displaystyle | d |}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d991a3d7e092278655e8bae004fee26d792608d1)
Properties of Pythagorean Quadruples
- The Pythagorean quadruple with the smallest product is .
![{\ displaystyle (1,2,2,3)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e3f82378068dfa7172fa3f85bdcf5c5c00f4b8f)
- Be with . Then:
![{\ displaystyle a ^ {2} + b ^ {2} + c ^ {2} = d ^ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a41c97edbec89d9eda011bb8314a977cec9d6f5)
![{\ displaystyle a, b, c, d \ in \ mathbb {N}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61c5a5becb6d0a0ae22de92b299bb32f10231c05)
- The product is always divisible by.
![{\ displaystyle abcd}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51b4d1adf9d72170e223b3e92b904eeebe836f84)
![12](https://wikimedia.org/api/rest_v1/media/math/render/svg/a522d3aa5812a136a69f06e1b909d809e849be39)
- There is no larger number that divides this product, because for the smallest Pythagorean quadruple (i.e. for ) . So there can be no greater number that divides the product.
![{\ displaystyle (a, b, c, d) = (1,2,2,3)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7189a36316c414cab4124f9c7d580c44af80a5a5)
![{\ displaystyle 1 \ times 2 \ times 2 \ times 3 = 12}](https://wikimedia.org/api/rest_v1/media/math/render/svg/789896680af9261b08a54fa9bb0d244661e817cc)
Generation of Pythagorean quadruples
- Be positive integers. Then the set of Pythagorean quadruples with odd can be generated as follows:
![{\ displaystyle (m, n, p, q)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13ae37d8a286a66ed6f4f8f4f062b7e0a775133f)
![a](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)
![{\ displaystyle {\ begin {aligned} a & = m ^ {2} + n ^ {2} -p ^ {2} -q ^ {2}, \\ b & = 2 (mq + np), \\ c & = 2 (nq-mp), \\ d & = m ^ {2} + n ^ {2} + p ^ {2} + q ^ {2} \ end {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ff6e486064e58c7a5d58d82ce95bb816c08ee4d)
- If the following eleven conditions also apply, then the set of primitive Pythagorean quadruples with odd can be generated.
![a](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)
![{\ displaystyle {\ begin {array} {lll} nq> mp, && m ^ {2} + n ^ {2}> p ^ {2} + q ^ {2}, \\ m \ geq 0, \; n \ geq 1, \; p \ geq 0, \; q \ geq 1, && m + p \ geq 1, \\ m + n + p + q \ equiv 1 {\ pmod {2}}, && {\ text { that is,}} m + n + p + q {\ text {is odd (so one or three values must be even numbers)}} \\ gcd (m ^ {2} + n ^ {2}, p ^ {2} + q ^ {2}, mq + np) = 1, && \\ m = 0 \; \ Longrightarrow \; q \ leq p, && p = 0 \; \ Longrightarrow \; n \ leq m \ end {array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d18cd6cb9c8d6bfb9056c7a53cb55703bea7bf36)
- All primitive Pythagorean quadruples thus satisfy the Diophantine equation , which is also called Lebesgue's identity :
![{\ displaystyle d ^ {2} = a ^ {2} + b ^ {2} + c ^ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9fe19213975ed949170191be9d3089968ff4eae)
![{\ displaystyle (m ^ {2} + n ^ {2} + p ^ {2} + q ^ {2}) ^ {2} = (m ^ {2} + n ^ {2} -p ^ {2 } -q ^ {2}) ^ {2} + (2mq + 2np) ^ {2} + (2nq-2mp) ^ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1eaa687e2d7a0f02e70d33ea08a6e4183579dc3f)
-
Example 1:
- Be and . Then all additional conditions are met and it is and and actually is a primitive Pythagorean quadruple.
![{\ displaystyle m: = 1, n: = 7, p: = 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50772a2e2f0bcf5118c1c0b55810d22b1a2b861f)
![{\ displaystyle q: = 5}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07d585276b6b7de138455d906ac6e697658a9108)
![{\ displaystyle a = 21, b = 38, c = 66}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0d034a2fce8bd769e1075a9df14259b3045149c)
![{\ displaystyle d = 79}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb9cdac35fa9c0ab590d3330cd66c3a6b3b59f8e)
![{\ displaystyle a ^ {2} + b ^ {2} + c ^ {2} = 21 ^ {2} + 38 ^ {2} + 66 ^ {2} = 79 ^ {2} = d ^ {2} \ quad (= 6241)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/720e4a4c96f66b76d420aa17cc288893f24bb83a)
-
Example 2:
- Be and . Then the additional condition is not fulfilled, but and because of it it is still a Pythagorean quadruple, but with .
![{\ displaystyle m: = 2, n: = 3, p: = 5}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67bcfc517aee8167420fbdcbe08f4de9136f3847)
![{\ displaystyle q: = 9}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a831d7f15e4b482c6f9dc8a5c5c92a786adc92d)
![{\ displaystyle m ^ {2} + n ^ {2} = 13 {\ stackrel {!} {>}} 106 = p ^ {2} + q ^ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09371a53737db63daccf5d74ef2bcb473a7f6a70)
![{\ displaystyle a = -93, b = 66, c = 34}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6cdc7de74814781204aabedfa5b985d1a317528)
![{\ displaystyle d = 119}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c50372ff1dcd7ebc67e26d6d47c0dd9c59427e9)
![{\ displaystyle a ^ {2} + b ^ {2} + c ^ {2} = (- 93) ^ {2} + 66 ^ {2} + 34 ^ {2} = 119 ^ {2} = d ^ {2} \ quad (= 14161)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ff7c2b355bf8016f7184cdf5a9c1ec523d63ad3)
![{\ displaystyle a = -93 <0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/953aae8667ce232f9b1ba0eb4d8c78696c74a202)
-
Example 3:
- Be and . Then is and and actually is . However, this Pythagorean quadruple is not primitive because and is the condition .
![{\ displaystyle m: = 1, n: = 3, p: = 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98c7a977bb7057a66e602d3c3f85d872f4be70a2)
![q: = 2](https://wikimedia.org/api/rest_v1/media/math/render/svg/db8433a5db9843041dcde4e955fa5392eec3bd61)
![{\ displaystyle a = 5, b = 10, c = 10}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5228c007b2ef7418ea3ee7fd7c402b3175ff278a)
![d = 15](https://wikimedia.org/api/rest_v1/media/math/render/svg/4121f3f61302443b56e3907e55ed6b3c565f9159)
![{\ displaystyle a ^ {2} + b ^ {2} + c ^ {2} = 5 ^ {2} + 10 ^ {2} + 10 ^ {2} = 15 ^ {2} = d ^ {2} \ quad (= 225)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7d69f8f159a4850b832be252ecb81efded5978e)
![{\ displaystyle \ operatorname {ggT} (a, b, c, d) = 5 \ not = 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a141f79fbf6a778524519eda1188ddeb9e74054)
![{\ displaystyle \ operatorname {ggT} (m ^ {2} + n ^ {2}, p ^ {2} + q ^ {2}, mq + np) = 5 \ not = 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/111ab1e29d76f690cbd3ce7050d168cd717c2d74)
- All Pythagorean quadruples (including the non-primitive) can be generated from two positive integers and as follows :
![a](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)
![b](https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3)
- Let the parity of and be different (so be either even and odd or odd and even). Furthermore, be a factor of with . Then:
![a](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)
![b](https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3)
![a](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)
![b](https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3)
![a](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)
![b](https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3)
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
![a ^ {2} + b ^ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14fc28103c5d2aa9276728469f82c9f415f4b257)
![{\ displaystyle p ^ {2} <a ^ {2} + b ^ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bffd771e321db8d6083c757013c4cbd11c5a4489)
-
and with![{\ displaystyle d = {\ frac {a ^ {2} + b ^ {2} + p ^ {2}} {2p}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89ed2a773ee9f476c247829b15e8848f53c1aee0)
-
Example:
- Be and . Then all the conditions are met and it is and (and it is ) and actually is .
![{\ displaystyle a: = 2, b: = 11}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5de46295e69dafdebf3049b247a26db3bfacbe5b)
![{\ displaystyle p: = 5}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55c6ac78373e08b5020952e7968093474060a6d1)
![{\ displaystyle c = 10}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4812139e0291998dddb93c5931f2696286a1ec91)
![d = 15](https://wikimedia.org/api/rest_v1/media/math/render/svg/4121f3f61302443b56e3907e55ed6b3c565f9159)
![{\ displaystyle p = dc}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90f31c0ef54861183e9b1a45aa5c08f10416f2d7)
![{\ displaystyle a ^ {2} + b ^ {2} + c ^ {2} = 2 ^ {2} + 11 ^ {2} + 10 ^ {2} = 15 ^ {2} = d ^ {2} \ quad (= 225)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19b75d9158e3cd886da246b98df5ef320f1cdeb7)
- Be and both even numbers. Also , be, and a factor of with . Then:
![a](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)
![b](https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3)
![{\ displaystyle l = {\ frac {a} {2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3277db8273b6e23c364219f96c457f6429ff587)
![{\ displaystyle m = {\ frac {b} {2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7af2f522d1d47e82bfea8f3cba7e4c64147c57ff)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![{\ displaystyle l ^ {2} + m ^ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe67315b8c93a20f924f41b9e85107a4b137c53a)
![{\ displaystyle n ^ {2} <l ^ {2} + m ^ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83dc99457b2ae33fdd8aed787c668ad6844ae490)
-
and
- This method generates all Pythagorean quadruples exactly once when and iterates over all pairs of natural numbers and iterates over all possible values for each pair.
![l](https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac)
![m](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
-
Example:
- Be and . Then all the conditions are met , and it is and and actually is .
![{\ displaystyle a: = 14, b: = 6}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39f1b8653aaded3935cbb660d3753fc28f186410)
![{\ displaystyle n: = 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4c70ddfd5feac6a0cab4892501418660a13848b)
![{\ displaystyle l = 7}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1c8a01c077ae742432b1ccd40b4bf7bd1c2be64)
![m = 3](https://wikimedia.org/api/rest_v1/media/math/render/svg/918bb386a7ca6891255b62ef91ccc022883f3809)
![{\ displaystyle c = 27}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d07046b39936487c5f996373639ac7491eb4c8e)
![{\ displaystyle d = 31}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e9c1c5b17ce8e2c1feb588baae9091bbbdd62e8)
![{\ displaystyle a ^ {2} + b ^ {2} + c ^ {2} = 14 ^ {2} + 6 ^ {2} + 27 ^ {2} = 31 ^ {2} = d ^ {2} \ quad (= 961)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/046485c81b827b256c26a561151a59fdcdbc83ac)
- There is no Pythagorean quadruple, in which more than one of the numbers , , is odd.
![a](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)
![b](https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3)
![c](https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455)
See also
Web links
Individual evidence
-
↑ To the spelling: In the current Duden - The large dictionary of the German language in ten volumes - ISBN 3-411-70360-1 the adjective "Pythagorean" is given in this spelling and the spelling "Pythagorean" is designated as an Austrian special form.
-
↑ a b Robert Spira : The Diophantine Equation x 2 + y 2 + z 2 = m 2 . The American Mathematical Monthly 69 (5), 1962, pp. 360-365 , accessed October 11, 2019 .
-
^ Raymond A. Beauregard, ER Suryanarayan: Pythagorean Boxes. Mathematics Magazine 74 (3), June 2001, pp. 222-227 , accessed October 11, 2019 .
-
↑ Des MacHale, Christian van den Bosch: Generalizing a result about Pythagorean triples. The Mathematical Gazette 96 (535), March 2012, pp. 91-96 , accessed October 11, 2019 .
-
^ Paul Oliverio: Self-Generating Pythagorean Quadruples and n -Tuples. Jefferson High School, Los Angeles, Dec. 1993, pp. 98-101 , accessed October 18, 2019 .
-
^ Robert Spira: The Diophantine Equation x 2 + y 2 + z 2 = m 2 , Theorem 2. The American Mathematical Monthly 69 (5), 1962, p. 362 , accessed October 11, 2019 .
-
↑ Pythagorean Quadruple. GeeksforGeeks - A computer science portal for geeks, accessed October 11, 2019 .
-
↑ Eric W. Weisstein : Lebesgue Identity. Wolfram MathWorld , accessed October 18, 2019 .
-
↑ Titu Andreescu, Dorin Andrica, Ion Cucurezeanu: An Introduction to Diophantine Equations: A Problem-Based Approach, Theorem 2.2.3. Birkhäuser, p. 79 , accessed on October 18, 2019 .