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Borromean coprimes
  • Given 3 positive integers, indicate whether they are Borromean coprimes.
  • ## Definition
  • 3 positive integers are called ***Borromean coprimes*** if both of the following are true:
  • - Their greatest common divisor is 1.
  • - The greatest common divisor of every pair is greater than 1.
  • In summary, the triple of integers is [coprime](https://en.m.wikipedia.org/wiki/Coprime_integers#Coprimality_in_sets), but removing any single integer leaves a pair of integers that are not coprime. The name is by analogy with [Borromean rings](https://en.m.wikipedia.org/wiki/Borromean_rings).
  • ## Input
  • - 3 positive integers.
  • - This may be as 3 separate inputs, or any ordered data structure.
  • - Your code must work for integers in any order (you must not assume that they are sorted).
  • - Your code must support input integers up to and including 127.
  • ## Output
  • - 1 of 2 distinct values to represent "True" and "False".
  • ## Examples
  • ### GCD not equal to 1 for the triple
  • - Input: `2, 4, 6`
  • - Output: `False`
  • The greatest common divisor of the triple is 2, so these are not Borromean coprimes.
  • ### GCD equal to 1 for a pair
  • - Input: `2, 3, 5`
  • - Output: `False`
  • The greatest common divisor of the pair `2, 3` is 1, so these are not Borromean coprimes.
  • ### Borromean coprimes
  • - Input: `6, 10, 15`
  • - Output: `True`
  • The greatest common divisors of each pair are:
  • - GCD(6, 10) = 2
  • - GCD(6, 15) = 3
  • - GCD(10, 15) = 5
  • The greatest common divisor of the triple is 1, and the greatest common divisor of every pair is greater than 1, so these are Borromean coprimes.
  • ## Non-golfed example code
  • Here is some Python code that meets the requirements of this challenge. The function `borromean_coprimes` returns `True` or `False`.
  • ```python
  • from math import gcd
  • def borromean_coprimes(x, y, z):
  • return (
  • coprime_triple(x, y, z)
  • and not coprime(x, y)
  • and not coprime(x, z)
  • and not coprime(y, z)
  • )
  • def coprime(x, y):
  • return gcd(x, y) == 1
  • def coprime_triple(x, y, z):
  • return gcd(x, y, z) == 1
  • ```
  • ## Test cases
  • Test cases are in the format `comma, separated, inputs : "output"`.
  • You may use any 2 distinct values instead of "True" and "False".
  • ```text
  • 1, 1, 1 : "False"
  • 1, 1, 2 : "False"
  • 1, 1, 3 : "False"
  • 1, 2, 2 : "False"
  • 1, 2, 3 : "False"
  • 2, 2, 2 : "False"
  • 2, 2, 3 : "False"
  • 2, 3, 3 : "False"
  • 2, 3, 4 : "False"
  • 2, 3, 5 : "False"
  • 2, 4, 5 : "False"
  • 2, 4, 6 : "False"
  • 127, 127, 127: "False"
  • 18, 33, 88 : "True"
  • 108, 20, 105 : "True"
  • 98, 30, 105 : "True"
  • 22, 36, 33 : "True"
  • 82, 30, 123 : "True"
  • 40, 55, 22 : "True"
  • 45, 12, 10 : "True"
  • 38, 57, 78 : "True"
  • 35, 84, 80 : "True"
  • 84, 33, 22 : "True"
  • 105, 54, 80 : "True"
  • 26, 96, 39 : "True"
  • 18, 26, 117 : "True"
  • 50, 75, 48 : "True"
  • 95, 76, 70 : "True"
  • 50, 96, 45 : "True"
  • 85, 34, 40 : "True"
  • 84, 104, 39 : "True"
  • 45, 72, 110 : "True"
  • 72, 68, 51 : "True"
  • 20, 105, 28 : "True"
  • 75, 102, 100 : "True"
  • 90, 105, 14 : "True"
  • 105, 110, 84 : "True"
  • 78, 70, 21 : "True"
  • 105, 96, 14 : "True"
  • 110, 120, 33 : "True"
  • 70, 84, 15 : "True"
  • 50, 6, 105 : "True"
  • 70, 21, 45 : "True"
  • 48, 70, 21 : "True"
  • 76, 18, 57 : "True"
  • 126, 77, 66 : "True"
  • 6, 88, 99 : "True"
  • 33, 77, 126 : "True"
  • 88, 72, 33 : "True"
  • 12, 63, 56 : "True"
  • 80, 36, 105 : "True"
  • 35, 110, 77 : "True"
  • 21, 14, 18 : "True"
  • 68, 85, 70 : "True"
  • 75, 108, 80 : "True"
  • 18, 21, 98 : "True"
  • 26, 36, 39 : "True"
  • 30, 98, 21 : "True"
  • 50, 15, 36 : "True"
  • 78, 51, 34 : "True"
  • 44, 98, 77 : "True"
  • 114, 105, 80 : "True"
  • 15, 10, 72 : "True"
  • 5, 91, 18 : "False"
  • 51, 41, 98 : "False"
  • 66, 78, 20 : "False"
  • 76, 18, 50 : "False"
  • 124, 105, 50 : "False"
  • 54, 1, 93 : "False"
  • 60, 41, 104 : "False"
  • 127, 62, 40 : "False"
  • 112, 101, 122 : "False"
  • 7, 12, 74 : "False"
  • 18, 95, 71 : "False"
  • 123, 74, 3 : "False"
  • 51, 79, 7 : "False"
  • 9, 67, 98 : "False"
  • 37, 6, 90 : "False"
  • 43, 1, 45 : "False"
  • 36, 14, 44 : "False"
  • 37, 1, 111 : "False"
  • 55, 89, 26 : "False"
  • 90, 53, 28 : "False"
  • 83, 12, 31 : "False"
  • 19, 112, 5 : "False"
  • 92, 19, 99 : "False"
  • 58, 59, 124 : "False"
  • 9, 106, 85 : "False"
  • 108, 108, 6 : "False"
  • 69, 31, 76 : "False"
  • 96, 6, 42 : "False"
  • 105, 47, 90 : "False"
  • 43, 22, 29 : "False"
  • 113, 19, 73 : "False"
  • 77, 103, 113 : "False"
  • 91, 89, 17 : "False"
  • 60, 16, 61 : "False"
  • 44, 87, 115 : "False"
  • 28, 80, 108 : "False"
  • 11, 116, 76 : "False"
  • 105, 79, 95 : "False"
  • 62, 80, 80 : "False"
  • 7, 60, 104 : "False"
  • 91, 106, 34 : "False"
  • 125, 105, 56 : "False"
  • 9, 74, 87 : "False"
  • 88, 68, 6 : "False"
  • 40, 17, 109 : "False"
  • 116, 83, 29 : "False"
  • 102, 32, 110 : "False"
  • 121, 20, 85 : "False"
  • 112, 44, 121 : "False"
  • 74, 102, 39 : "False"
  • ```
  • ## Scoring
  • This is a [code golf challenge]. Your score is the number of bytes in your code.
  • > Explanations are optional, but I'm more likely to upvote answers that have one.
  • [code golf challenge]: https://codegolf.codidact.com/categories/49/tags/4274 "The code-golf tag"
  • Given 3 positive integers, indicate whether they are Borromean coprimes.
  • ## Definition
  • 3 positive integers are called ***Borromean coprimes*** if both of the following are true:
  • - Their greatest common divisor is 1.
  • - The greatest common divisor of every pair is greater than 1.
  • In summary, the triple of integers is [coprime](https://en.m.wikipedia.org/wiki/Coprime_integers#Coprimality_in_sets), but removing any single integer leaves a pair of integers that are not coprime. The name is by analogy with [Borromean rings](https://en.m.wikipedia.org/wiki/Borromean_rings).
  • ## Input
  • - 3 positive integers.
  • - This may be as 3 separate inputs, or any ordered data structure.
  • - Your code must work for integers in any order (you must not assume that they are sorted).
  • - Your code must support input integers up to and including 127.
  • ## Output
  • - 1 of 2 distinct values to represent "True" and "False".
  • ## Examples
  • ### GCD not equal to 1 for the triple
  • - Input: `2, 4, 6`
  • - Output: `False`
  • The greatest common divisor of the triple is 2, so these are not Borromean coprimes.
  • ### GCD equal to 1 for a pair
  • - Input: `2, 3, 5`
  • - Output: `False`
  • The greatest common divisor of the pair `2, 3` is 1, so these are not Borromean coprimes.
  • ### Borromean coprimes
  • - Input: `6, 10, 15`
  • - Output: `True`
  • The greatest common divisors of each pair are:
  • - GCD(6, 10) = 2
  • - GCD(6, 15) = 3
  • - GCD(10, 15) = 5
  • The greatest common divisor of the triple is 1, and the greatest common divisor of every pair is greater than 1, so these are Borromean coprimes.
  • ## Non-golfed example code
  • Here is some Python code that meets the requirements of this challenge. The function `borromean_coprimes` returns `True` or `False`.
  • ```python
  • from math import gcd
  • def borromean_coprimes(x, y, z):
  • return (
  • coprime_triple(x, y, z)
  • and not coprime(x, y)
  • and not coprime(x, z)
  • and not coprime(y, z)
  • )
  • def coprime(x, y):
  • return gcd(x, y) == 1
  • def coprime_triple(x, y, z):
  • return gcd(x, y, z) == 1
  • ```
  • ## Test cases
  • Test cases are in the format `comma, separated, inputs : "output"`.
  • You may use any 2 distinct values instead of "True" and "False".
  • ```text
  • 1, 1, 1 : "False"
  • 1, 1, 2 : "False"
  • 1, 1, 3 : "False"
  • 1, 2, 2 : "False"
  • 1, 2, 3 : "False"
  • 2, 2, 2 : "False"
  • 2, 2, 3 : "False"
  • 2, 3, 3 : "False"
  • 2, 3, 4 : "False"
  • 2, 3, 5 : "False"
  • 2, 4, 5 : "False"
  • 2, 4, 6 : "False"
  • 127, 127, 127 : "False"
  • 18, 33, 88 : "True"
  • 108, 20, 105 : "True"
  • 98, 30, 105 : "True"
  • 22, 36, 33 : "True"
  • 82, 30, 123 : "True"
  • 40, 55, 22 : "True"
  • 45, 12, 10 : "True"
  • 38, 57, 78 : "True"
  • 35, 84, 80 : "True"
  • 84, 33, 22 : "True"
  • 105, 54, 80 : "True"
  • 26, 96, 39 : "True"
  • 18, 26, 117 : "True"
  • 50, 75, 48 : "True"
  • 95, 76, 70 : "True"
  • 50, 96, 45 : "True"
  • 85, 34, 40 : "True"
  • 84, 104, 39 : "True"
  • 45, 72, 110 : "True"
  • 72, 68, 51 : "True"
  • 20, 105, 28 : "True"
  • 75, 102, 100 : "True"
  • 90, 105, 14 : "True"
  • 105, 110, 84 : "True"
  • 78, 70, 21 : "True"
  • 105, 96, 14 : "True"
  • 110, 120, 33 : "True"
  • 70, 84, 15 : "True"
  • 50, 6, 105 : "True"
  • 70, 21, 45 : "True"
  • 48, 70, 21 : "True"
  • 76, 18, 57 : "True"
  • 126, 77, 66 : "True"
  • 6, 88, 99 : "True"
  • 33, 77, 126 : "True"
  • 88, 72, 33 : "True"
  • 12, 63, 56 : "True"
  • 80, 36, 105 : "True"
  • 35, 110, 77 : "True"
  • 21, 14, 18 : "True"
  • 68, 85, 70 : "True"
  • 75, 108, 80 : "True"
  • 18, 21, 98 : "True"
  • 26, 36, 39 : "True"
  • 30, 98, 21 : "True"
  • 50, 15, 36 : "True"
  • 78, 51, 34 : "True"
  • 44, 98, 77 : "True"
  • 114, 105, 80 : "True"
  • 15, 10, 72 : "True"
  • 5, 91, 18 : "False"
  • 51, 41, 98 : "False"
  • 66, 78, 20 : "False"
  • 76, 18, 50 : "False"
  • 124, 105, 50 : "False"
  • 54, 1, 93 : "False"
  • 60, 41, 104 : "False"
  • 127, 62, 40 : "False"
  • 112, 101, 122 : "False"
  • 7, 12, 74 : "False"
  • 18, 95, 71 : "False"
  • 123, 74, 3 : "False"
  • 51, 79, 7 : "False"
  • 9, 67, 98 : "False"
  • 37, 6, 90 : "False"
  • 43, 1, 45 : "False"
  • 36, 14, 44 : "False"
  • 37, 1, 111 : "False"
  • 55, 89, 26 : "False"
  • 90, 53, 28 : "False"
  • 83, 12, 31 : "False"
  • 19, 112, 5 : "False"
  • 92, 19, 99 : "False"
  • 58, 59, 124 : "False"
  • 9, 106, 85 : "False"
  • 108, 108, 6 : "False"
  • 69, 31, 76 : "False"
  • 96, 6, 42 : "False"
  • 105, 47, 90 : "False"
  • 43, 22, 29 : "False"
  • 113, 19, 73 : "False"
  • 77, 103, 113 : "False"
  • 91, 89, 17 : "False"
  • 60, 16, 61 : "False"
  • 44, 87, 115 : "False"
  • 28, 80, 108 : "False"
  • 11, 116, 76 : "False"
  • 105, 79, 95 : "False"
  • 62, 80, 80 : "False"
  • 7, 60, 104 : "False"
  • 91, 106, 34 : "False"
  • 125, 105, 56 : "False"
  • 9, 74, 87 : "False"
  • 88, 68, 6 : "False"
  • 40, 17, 109 : "False"
  • 116, 83, 29 : "False"
  • 102, 32, 110 : "False"
  • 121, 20, 85 : "False"
  • 112, 44, 121 : "False"
  • 74, 102, 39 : "False"
  • ```
  • ## Scoring
  • This is a [code golf challenge]. Your score is the number of bytes in your code.
  • > Explanations are optional, but I'm more likely to upvote answers that have one.
  • [code golf challenge]: https://codegolf.codidact.com/categories/49/tags/4274 "The code-golf tag"
code-golf math
code-golf math

Suggested about 1 month ago by celtschk‭