Monthly Archives: March 2016

Integers w/ 4 different partitions into 3 parts w/ the same product

    Partition (number theory) https://en.wikipedia.org/wiki/Partition_(number_theory)     Find integers     that have 4 different partitions into 3 parts with the same product and with the same product,                        … Continue reading

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fractional part of (6*sqrt(6) + 14)

            Prove that if    ,   then                                

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The square of a Fibonacci number

    Show that is the square of a Fibonacci number                                    

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Fibonacci numbers | is 5 F^2_{2k} + 4 always a square number?

    Is     always a square number? Also, is     always a square number?                                    

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Fibonacci numbers | F(n) < x < F(n+1) < y < F(n+2)

    If   then show that     is never a Fibonacci number                                    

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Fibonacci numbers – Identity

  Establish the identity:                              

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Alphametric puzzles — Part 1

  Definition:   An alphametic puzzle is an arithmetic problem involving words where there is a one-to-one mapping between letters and digits that makes the arithmetic equation true.   (1) in base 8,     is a prime number Solved … Continue reading

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Heron triangles & half-angle formulae

  Prove that If the sides of a triangle are in arithmetic progression if and only if the cotangents of its half-angles ,    ,    are also in arithmetic progression   Also,   if and only if       … Continue reading

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a^2 + b^2 + c^2 = d^2

    Prove that the product of the positive integers a and b is even if and only if there exist positive integers c and d such that                         … Continue reading

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(2*k + 1)^2 * T(n) + T(k) is a triangular number

    if     is the n-th triangular number then   is a triangular number     Hence,                                    

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