Relatively Prime Pythagorean TriplesDate: 09/13/97 at 18:23:22 From: Ben Love Subject: Relatively Prime Pythagorean Triplets A relatively prime Pythagorean triplet (RPPT) (in case you know it by other names) is a triplet such as 3,4,5 that cannot be reduced. Ex. 3,4,5 cannot be reduced. Ex. 6,8,10 CAN be reduced (by 2) Anyway, I found using a computer that whenever A is a positive prime number, then B is even and C = B + 1! Another interesting fact is that so far I have only been able to find exactly one RPPT for each prime from 1 to 100 except 2. 2, as far as I can see, doesn't work at all. I can't say I've proven that there is no other triplet with 3, but I have found that 4 is the ONLY number from 1 to 10,000 that works with 3. The third thing I've found is that if you take each prime from 3 to 100 and then find the difference between the "B"s (ex: 3 _4_ 5 and 5 _12_ 13 -- 12 - 4 = 8) all of the differences except the first and second are divisible by twelve. This brings me to my first question: Why is that? Or is that just a coincidence? Second, if that is true, how can I prove that C - B = 1 (when A is a pos. prime and B and C are pos. whole numbers)? I have been working on this and cannot figure it out. Here is a chart that I have printed up: The first three columns represent A, B, and C in that order. The fourth is the difference between each prime (A). The fifth is the difference between the other numbers (B). The last is what you get when you divide B by 12. ------ Copied from file rppt.prf -------- RPPT = Relatively Prime Pythagorean Triplet A triplet is relatively prime if the triplet cannot be reduced. Ex: 3,4,5 cannot be reduced. Ex: 6,8,10 CAN be reduced (by 2) so it is not. These are all RPPTs. A B C CC Diff. A B B / 12 --- --- --- ------ --- --- --- Here is one: 3 4 5 25 Difference: 3 4 0 Here is one: 5 12 13 169 Difference: 2 8 0 Here is one: 7 24 25 625 Difference: 2 12 1 Here is one: 11 60 61 3721 Difference: 4 36 3 ---------- Here is one: 13 84 85 7225 Difference: 2 24 2 Here is one: 17 144 145 21025 Difference: 4 60 5 10 primes Here is one: 19 180 181 32761 Difference: 2 36 3 Here is one: 23 264 265 70225 Difference: 4 84 7 in a row? Here is one: 29 420 421 177241 Difference: 6 156 13 Here is one: 31 480 481 231361 Difference: 2 60 5 Why? Here is one: 37 684 685 469225 Difference: 6 204 17 Here is one: 41 840 841 707281 Difference: 4 156 13 Here is one: 43 924 925 855625 Difference: 2 84 7 ---------- Here is one: 47 1104 1105 1221025 Difference: 4 180 15 Here is one: 53 1404 1405 1974025 Difference: 6 300 25 Here is one: 59 1740 1741 3031081 Difference: 6 336 28 Here is one: 61 1860 1861 3463321 Difference: 2 120 10 Here is one: 67 2244 2245 5040025 Difference: 6 384 32 Here is one: 71 2520 2521 6355441 Difference: 4 276 23 Here is one: 73 2664 2665 7102225 Difference: 2 144 12 Here is one: 79 3120 3121 9740641 Difference: 6 456 38 Here is one: 83 3444 3445 11868025 Difference: 4 324 27 Here is one: 89 3960 3961 15689521 Difference: 6 516 43 Here is one: 97 4704 4705 22137025 Difference: 8 744 62 Prove: When A is a pos. prime and B,C are pos. whole numbers, C-B = 1 B = Sqrt((CC)-(AA)) C = Sqrt((AA)+(BB)) C - B = Sqrt((AA)+(BB)) - Sqrt((CC)-(AA)) If C - B = 1 then B = (AA)-1 AND (AA) = 2B + 1 A B A B C 45 degrees +----+---+ +----+---+ | / A | AA | /| B| / \ | | / | |/ | +/ CC \|A |/ +----+---+ |\ / + -------------B B | | BB| A| \ / |B /| +----+---+ +---+----+ / | B A / | ------- Copied from file rppt.prf ------- Sqrt = Sq. Root AA, BB, CC, etc. = A squared, B squared, C squared, etc. The first two diagrams show the Pythagorean Proof. The third diagram is missing a curve. The curve is a hyperbola that touches the 45 degree line exactly once. It is a representation of a number A where it is a prime. The line represents the points where C = B + 1. The only point where A touches the line is the RPPT. The other places are where the triplet is not all whole numbers. The equation for the hyperbola is C = sqrt(BB+AA). This is what my dad and I have been able to come up with. Please help us prove C - B = 1. Thanks, Ben Love Date: 09/13/97 at 19:33:51 From: Doctor Anthony Subject: Re: Relatively Prime Pythagorean Triplets The fact that C-B = 1 can be proved quite quickly if you look at the formula for generating Pythagorean triples. If we require integer values of x, y, z for which x^2 + y^2 = z^2 then we can use the following generators with a, b taking any integer values whatsoever provided a > b. x = a^2 - b^2 y = 2ab z = a^2 + b^2 since x^2 + y^2 = a^4 - 2a^2.b^2 + b^4 + 4a^2.b^2 = a^4 + 2a^2.b^2 + b^4 = (a^2 + b^2)^2 = z^2 For example if a=2, b=1 produces x = 4 - 1 = 3 y = 2 x 2 x 1 = 4 z = 4 + 1 = 5 A prime number has no factors other than 1 and itself. Now note that x = (a-b)(a+b) so if x is prime the factor (a-b) MUST be 1. Also z-y = a^2 + b^2 - 2ab = (a-b)^2 = 1^2 = 1. So if x is prime then z and y MUST differ by 1. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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