Back around 1987, I was in an undergraduate number theory course, and I was trying to solve this silly problem as a project for the class. I came across Dickson's fantastic History of the Theory of Numbers, and found the chapter, Properties of the digits of numbers, that I thought might be helpful. In that chapter I found this:
P. W. Flood proved that 64 is the only square the sum of whose digits less unity and product plus unity are squares.
I don't recall at the time if I checked this fact (it is not true: 187489 is another such square, and my hunch is that there are infinitely many such, even if you exclude squares with zero digits), but I was very interested in seeing this proof. The reference given in Dickson is
Math. Quest. Educ. Times, 7, 1867, 30.
and I was not able to get my hands on whatever this was.
Recently (August 2007), I was perusing Dickson again, and I was reminded of this Flood entry. Thanks to techological developments over the last twenty years, I was able, almost accidentally, to find a scan of the actual proof on Google books (it's on page 30). Amazing!
However, the proof is, as expected, simply a quasi-proof that 64 is the only two digit square with this property. What the heck? There are (count 'em) six two digit square numbers; why in the world would you go to the trouble of proving that 64 was the only one with this property?
But, let me give you the proof, just for fun.
Suppose the number has digits x and y so 10x+y is a square, x+y-1 is a square, and xy+1 is a square. Take x=y+2, then xy+1=(y+1)2, and we have 11y+20 is a square, and 2y+1 is a square. Assume y = 2p2+2p, then 2y+1=(2p+1)2, and 11y+20 = 22p2+22p+20 is a square. This last expression will ("evidently") be a square if p=1; this gives 64 as the number. (Flood actually has the final sentence, "If fractional values were admissable, this last expression would admit of an indefinite number of solutions, which may be obtained by well-known methods.")
What the heck was that? Note particularly the "take x=y+2" bit. This by itself nails the number down to 64. So, this "proof" doesn't even prove that 64 is the only two digit square with the property, merely that 64 has the property!
So Dickson is actually incorrect. Anyone maintaining an errata collection for Dickson?
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