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wu :: forums    riddles    medium (Moderators: SMQ, Grimbal, Icarus, Eigenray, ThudnBlunder, william wu, towr)    Integer triangle with 120 degrees « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: Integer triangle with 120 degrees  (Read 2889 times) Christine Full Member     Posts: 159 Integer triangle with 120 degrees   « on: Apr 23rd, 2013, 10:35am » Quote Modify This type of triangle can be generated by   a = m2 - n2   b = 2mn + n2   c = m2  + mn + n2     we note   a2  + ab + b2  = c2    If m = 2, n = 1 ---> a = 3, b = 5, c = 7   the smallest triangle with an angle of 120 degrees having all 3 sides with prime numbers.   Is it the only triangle with the three sides being prime numbers? If yes, how to prove it? IP Logged towr wu::riddles Moderator Uberpuzzler Some people are average, some are just mean.     Gender: Posts: 13730 Re: Integer triangle with 120 degrees   « Reply #1 on: Apr 23rd, 2013, 10:41am » Quote Modify Well, n has to be 1, otherwise b can't be prime (n divides 2mn + n^2). And m-n has to be 1, because otherwise a isn't prime (m-n divides m^2-n^2). So therefore m=2, n=1 is the only solution with prime numbers.   Provided all such triangles can be generated in this way. « Last Edit: Apr 23rd, 2013, 10:43am by towr » IP Logged Wikipedia, Google, Mathworld, Integer sequence DB Christine Full Member     Posts: 159 Re: Integer triangle with 120 degrees   « Reply #2 on: Apr 23rd, 2013, 11:13am » Quote Modify on Apr 23rd, 2013, 10:41am, towr wrote: Well, n has to be 1, otherwise b can't be prime (n divides 2mn + n^2). And m-n has to be 1, because otherwise a isn't prime (m-n divides m^2-n^2). So therefore m=2, n=1 is the only solution with prime numbers.   Provided all such triangles can be generated in this way.   Can n be a prime number > 2 ? « Last Edit: Apr 23rd, 2013, 12:04pm by Christine » IP Logged Pages: 1  Reply Notify of replies Send Topic Print Forum Jump: ----------------------------- riddles -----------------------------  - easy => medium   - hard   - what am i   - what happened   - microsoft   - cs   - putnam exam (pure math)   - suggestions, help, and FAQ   - general problem-solving / chatting / whatever ----------------------------- general -----------------------------  - guestbook   - truth   - complex analysis   - wanted   - psychology   - chinese « Previous topic | Next topic »

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