What is the shortest possible other leg? Conditions in problem statement.
This problem's question: What is the shortest possible other leg? The longer leg of the primitive right triangle is factorial(100) (that is, 100!). The hypotenuse is longer than factorial(100).
This problem was solved by an exhaustive computer search.
The numbers involved are over 150 decimal digits long. This is a computer problem for most people.
The requested form of the answer, because the answer is so large, is , where means round to closest integer and to round to the even integer.
I offer some starter information. See Pythagorean triple . The value, factorial(100), factors as {{2,97},{3,48},{5,24},{7,16},{11,9},{13,7},{17,5},{19,5},{23,4},{29,3},{31,3},{37,2},{41,2},{43,2},{47,2},{53,1},{59,1},{61,1},{67,1},{71,1},{73,1},{79,1},{83,1},{89,1},{97,1}} where the left hand side of each pair is the prime and the right hand side of each pair is the power. The length of the longer leg in the same form as requested for the answer is . This is a thin triangle.
*Credit: * this problem was developed from my analysis in answering These Triangles Broke My Calculator .
The answer is 15223616.
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1 solution
The factors using the primes to their respective powers are: {79228162514264337593543950336, 79766443076872509863361, 59604644775390625, 33232930569601, 2357947691, 62748517, 1419857, 2476099, 279841, 24389, 29791, 1369, 1681, 1849, 2209, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97}.
Then, a search across all subsets of the powers of odd to find the smallest absolute value for m 2 − n 2 gives the answer.
The a set of the two sets of primes for the lowest difference (there might be others, I did not look) is {{2, 7, 13, 23, 29, 31, 43, 47, 73, 79, 89, 97}, {3, 5, 11, 17, 19, 37, 41, 53, 59, 61, 67, 71, 83}}.
Oh, that number is: 172250630912249478982342423751061720393048233093768282315878289049186356044734338122015520064583399044885434967726450679911846919325987487961487948872361.
The reason for looking for the narrowest triangle rather than the most isosceles triangle is that that problem is much simpler: n = 2 9 6 and m = 2 9 7 1 0 0 ! .