Two squares and powers of 50

1² + 7² = 50

 48² + 14² = 50²

 146² + 322² = 50³

 2108² + 1344² = 50⁴

 11516² + 13412² = 50⁵

 82368² + 94024² = 50⁶
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Result follows from a simple rule.

If:  u² + 49v² = 50^a

Then: (u - 49v)² + 49(u + v)² = 50^a+1

Which is just using my BQD Iterator. And u or v can be positive or negative, so I selected to eliminate trivial solutions. Let's divide off shared factors where can, and highlight a pattern:

1² + 7² = 2(5²)

 24² + 7² = 5⁴

 73² + 161² = 2(5⁶)

 527² + 336² = 5⁸

 2879² + 3353² = 2(5¹⁰)

 10296² + 11753² = 5¹²


So yeah, found myself curious again, where have already shown with powers of 10 and powers of 5 from those, and realized hey, can do 50 as well. And I do find it interesting looking back at overlap between lists, without seeing any different values.

The more general reality is that there always exists nonzero x and y, such that for an integer n equal to 0 or higher, and an integer m equal to 3 or higher:

x² + (m-1)y² = mⁿ⁺¹

Where yes, can extrapolate from rule shown above, as actually I used the more general form:

If:  u² + (m-1)v² = m^a

Then: (u - (m-1)v)² + (m-1)(u + v)² = m^a+1

My BQD Iterator is tool being used to get those values. And on this subject can see more with this post from 2014.

And am endlessly fascinated by it, even as I use it in what I see as marketing ways. So here use it to do sums of two squares to powers but have also used it to do arbitrary sums of squares to a square, as it shows that squares and exponents have this deep relationship.

These relationships control integers, whether you know of them, or believe in them, or not.

The math does not care. That infinite coldness of absolute knowledge has bothered me at times, but now I simply appreciate it, and all it can do.

The math has infinite knowledge. And the math can do so much where I like what has been shown to me. But I DO appreciate the knowledge.

The math knows.


James Harris