Interesting thing I've noted

[karpmage]

2^2-1 = 2^2-1 = 1^2+2
2^3-1 = 3^2-2 = 2^2+3
2^5-1 = 6^2-5 = 5^2+6
2^13-1 = 91^2-90 = 90^2+91

I realize that x^2-(x-1)==(x-1)^2+(x)

2,3,5 and 13 are all the powers of mersenne primes, and are Fibonacci numbers as well.
It'd be interesting to see what's the next power of 2 that satisfies this equation.

 

[Norwegian]

Hi there, you have observed some cases of powers of 2 being almost in the middle of consecutive squares, and you ask about the next occurrence. A quick computer search reveals no such cases with exponent less than 1000. So unless either I or my computer made a mistake, which may very well happen, you may want to look for other patterns instead.

What happened to your post about sums and differences of two powers? I had written a reply, including this http://oeis.org/A074981http://, and the relation 30=83²-19³, and some more, and then I couldn't post it.

 

[karpmage]

Sorry, I posted this to another maths forum as well, and it turns out there are no other numbers. Here's the link I was sent:
http://oeis.org/A215797

 

[karpmage]

Hi there, you have observed some cases of powers of 2 being almost in the middle of consecutive squares, and you ask about the next occurrence. A quick computer search reveals no such cases with exponent less than 1000. So unless either I or my computer made a mistake, which may very well happen, you may want to look for other patterns instead.

What happened to your post about sums and differences of two powers? I had written a reply, including this http://oeis.org/A074981http://, and the relation 30=83²-19³, and some more, and then I couldn't post it.

My last post violated forum rules, I am going to repost a revised version of it. Didn't see your reply before they deleted it, unfortunately.