## Square numbers split up

Several square numbers can be split up into other square numbers. Examples: 49 split up as 4 and 9, 169 split up as 16 and 9, 1225 split up as 1 and 225, 1681 split up as 16 and 81, 4225 split up as 4 and 225. Are there more such numbers?

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A method for finding some examples:
Let’s look for cases where second square is a four-digit number n², and the first square is m², and the ”entire” number k². Then k² = 10000m² + n², or

(100m)² + n² = k²

Now we look for p, q such that 100m = 2pq, n = p² – q² so that k = p² + q². (This is the usual parametrization of Pythagorean numbers). Then

50m = pq and n = (p + q)(p – q)

Further let’s take p = q + 1. (I’m just looking for some example[s], though more can be looked in like manner.) Then

5²*2m = (q + 1)q and n = 2q + 1.

To satisfy the first equation an obvious possibility is q = 24, giving m = 12, n = 49.

1442401 = (25² + 24²)² = 1201² >>> 144 = 12² , 2401 = 49²

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