- #1
ramsey2879
- 841
- 3
I have a new conjecture re triangular numbers that I think is fascinating.
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].
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].
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