Monthly Archives: April 2015

Integer which is a palindrome in precisely 3 consecutive bases (excluding base 10)

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Find in base-6 a 9-digit square of the form AAAAAAXYZ

  Hint :   it is the square of a number whose central triad is XYZ Answer:   333333521   base-6     David found:                      

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Num3er 952

      is the sum of the cubes of its digits plus the product of its digits:     Can you find other examples?   Paul found:                         … Continue reading

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Integers n such that n(n+1) = (n+2) (mod n+3)

                                                          Solve the following: (1)     (2)   where     is not necessarily     (3)                            

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When the product of two positive integers is even

    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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Perfect squares formed by the last d-digits exceeding the first d-digits by 1

  that is,         (Concatenation of 6099 and 6099+1)                                  

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Inequality: 2^(b+c) + 2^(c+a) + 2^(a+b) < 2^(a+b+c+1) + 1

  Prove that                                                    

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Equation: x^(a+b) + y = (x^a)(y^b)

  Solve when   a,   b,   x,   y   are in                                  

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Oblong numbers| When is the sum of two oblong numbers oblong?

    Oblong numbers are of the form ,     where     is the n-th triangular number. An integer of the form     is called a Triangular number.   Here are all the Oblong numbers   with   … Continue reading

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Polygonal number relation| m-gonal = T(n) + (m-3)*T(n-1)

    An integer of the form     is called a Triangular number. The sum of two consecutive Triangle numbers is always a square number So,     Every pentagonal number can be written as a sum of a triangular … Continue reading

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