I have a new conjecture re triangular numbers that I think is fascinating.(adsbygoogle = window.adsbygoogle || []).push({});

Conjecture

For any two integers [tex]a[/tex] and [tex]b[/tex] such that [tex]ab[/tex] is a triangular number, then there is an integer [tex]c[/tex] such that [tex]a^2 + ac[/tex] and [tex]b^2 + bc[/tex] are both triangular numbers. Further, [tex](6b-a+2c)*b[/tex] and [tex](6b-a+2c)*(6b-a+3c)[/tex] are also triangular numbers so this property is recursive.

an interesting set of such recursive series is

0,1,6,35,204 ...(c = 0)

0,2,14,84,492...(c = 1)

0,3,22,133,780..(c = 2)

...

where the differences between any two sucessive terms of the [tex]i[/tex]th columm form the recursive series [tex]0,1,8,49,288..(6*n_{(i-1)}-n_{(i-2)}+2)[/tex].

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# New Conjecture

Loading...

Similar Threads for Conjecture | Date |
---|---|

Our Old Friend, the Twin Primes Conjecture | Oct 23, 2014 |

Beal Conjecture - Demonstration | Dec 11, 2013 |

Proof of the twin primes conjecture | Jun 29, 2013 |

Some ideas concerning the Goldbach conjecture | Dec 25, 2012 |

A Conjecture on the Collatz Conjecture | Nov 16, 2012 |

**Physics Forums - The Fusion of Science and Community**