# Category Archives: Uncategorized

## Triangular & Second pentagonal numbers

Suppose the positive integers     satisfy   ………. (1) Or     is a Triangular number  [     ] numbers of the form     are Second pentagonal numbers.   Equation   (1)   gives us the … Continue reading

## Alphametic puzzles — Part 2

(1)   are positive integers in arithmetic sequence such that ,     and     Find     (2) Substitute each of the letters with a different decimal digit from 0 to 9, to satisfy the following system of simultaneous … Continue reading

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## Positive integers (a,b,c) in AP; a+b, b+c, c+a are squares

are in arithmetic progression   Here are some solutions:

## Pythagorean triples (x^2 – 1, y^2 – 1, z^2 – 1)

To find a Pythagorean triangle with sides, where     are integers.   Here are two examples,

## Concatenation: 2016 || x || 2016 = y^3

Previous blog:   Concatenation: 2016 || x || 2016 = y^2   Paul found: ………………..     Pipo found: A few more, 2016 appear every 2500 starting at 856                     … Continue reading

## a^n + b^n + c^n = x^n + y^n + z^n, n = 2, 4

Find six distinct positive integers     such that     for example,     …..     Paul found:   …     …     …     …     …     …     … Continue reading

## Open question: Positive integers as a sum of squares and cubes?

Every positive integer is a sum of four squares (Lagrange’s Theorem) It is known that every positive integer is a sum of no more than 9 positive cubes, and that every “sufficiently large” integer is a sum of … Continue reading

## 123456789+987654321

777777   has   48 divisors: 1   3   7   11   13   21   33   37   39   49   77   91   111   143   147   231   259   … Continue reading