Simply interesting across centuries

Will admit one of my more gratifying finds can be shown with some really simple examples, where luckily noticed that both Ramanujan and Euler had shown interest in the area.

Here are some expressions:

1² + 7 = 2³

Next : 3² + 7 = 2⁴

And next is: 5² + 7 = 2⁵

And then: 1² + 7(3)² = 2⁶

One more: 11² + 7 = 2⁷

Notice that 7 is bare, in all but one case.

And according to MathWorld, Ramanujan has a result called Ramanujan's square equation:

x² + 7 = 2ⁿ

Which they say only has solutions for n = 3, 4, 5, 7 and 15.

And I've shown the n = 3, 4, 5 and 7 cases, but am not interested in iterating 8 more times to get the last one.

And also on MathWorld I found that Euler had an interest in this area as well with his own solution which was NOT using my BQD Iterator. Will admit have no clue how he found it, as is something more general.

Where I explain more in a previous post where I note these as validating historical connections.

My methods generalize to allow you to solve a much larger variety, ok an infinite variety of similar. Will admit that being able to explain with something that Euler probably didn't know and that Ramanujan probably didn't know in an area where both showed an interest feels weird. Am doing better with it now as time has passed. You just kind of feel good after a bit and lose most of the overawed feeling.

That celebrity aspect does make me ponder a bit, but will admit is good fun.


James Harris