Monthly Archives: August 2014

Puzzle: Triangular numbers, n+p and T(n)+T(p)

      Gary (republicofmath) found :       (2)   solve analytically               Advertisements

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Patterns | (x + y + z)^n – (x^n + y^n + z^n); n = 2,3,5,7

      in     exponents being the prime numbers   2, 3, 5, 7 :     Find out whether the pattern continues with   n = 11, 13                   … Continue reading

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N = a^2 + b^2 + (a b) = c^2 + d^2 + (c d)

  Find all integers     below 10000 which are expressible in two ways: and   For example,     Let’s take the first few terms of the sequence 91, 133, 273, 343, 553, 651, 703, 871, 931, 1057, 1261, … Continue reading

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x^n + y^n + z^n = u^n + v^n + w^n; n = 1,3

  If     and       Is the statement     true   ?   then     then   But if   that is ,     ,     ,     ,     Since then       when      … Continue reading

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x^4 – y^4 = z^3

                                                                 ——————————————     Contributor: David Radcliffe                                                              ——————————————                      

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N = x^3 + y^2 = (a^3 + b^2)(c^2 + d^2)

    To find     such that where     are nonzero integers   for example,   (1)   Find more solutions   Challenging question: (2)   Generalize this                    

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(sin(x) + sin(2 x) + … + sin(n x))/(cos(x) + cos(2 x) + … + cos(n x))

  Prove that                        

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Patterns | (x + y)^p – x^p – y^p where p is a prime number

  Let   ,        ,              p is a prime number           but so so   So   but so,       Note the pattern:   Determine                                                                    —————————————— … Continue reading

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(n, n+1) Higher Sum and Product

  Among the distinct pairs of numbers whose sum is   ,   the greatest possible product is   Consider   ,   find the distinct ways of achieving a higher product with two positive integers   Here are the … Continue reading

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Sum of six 5-th powers

  So                    

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