# [seqfan] Relationship between Bézout coefficients and powers of 2 and 3

Joe Slater seqfan at slatermold.com
Sun Mar 19 13:17:19 CET 2017

```I came across what seems to be an interesting link between the Bézout
coefficients (a,b) in the expression ax - by = c when c is set to 1 and x
and y are set to powers of 2 and 3.

If we set b = 3 and a= (1,2,4 ...) then the value of y seems to take the
values 1, 5, 21, 85 ..., which is A002450

If we set b = 9 and a= (1,2,4 ...) then the value of y seems to take the
values1, 3, 7, 71, 199 ..., which is A113841

If we set b= 27 and a= (1,2,4 ...) then the value of y seems to take the
values 1, 5, 13, 45, 109..., which is A113835

In all these cases the value of y sometimes takes its value multiplied by a
power of two, but the power is successively reduced until y takes its next
value from the series. I.e., y=28, y=14, y=7, followed by y= 71.

I haven't gone any further than this, and I'm pretty sure that I've just
imported the Euclidean Algorithm through the back door, but I thought it
was an interesting set of correspondences.
```