Sums of Fourth Powers

Another fun John Cook post has sent me off into playing with numbers and Perl 6.

He gives a footnote saying that Euler discovered 635318657 = 158^4 + 59^4 = 134^4 + 133^4 and that this was the smallest number known to be the sum of two fourth powers in two ways. It seems odd now to think of such questions being unresolved. Today we’d ask Hardy “What do you mean 635318657 is the smallest known example? Why didn’t you write a little program to find out whether it really is the smallest?”

Perl 6 has features that make that fun to explore, so let’s go to it! Here was my first attempt:

my @fourth-powers = (1..*).map(* ** 4) ...^ * > 635318657;
my %counts;
for @fourth-powers X+ @fourth-powers -> $sum {
    %counts{$sum}++;
}

for %counts.keys.grep({ %counts{$_} > 2 }) -> $sum {
    say "$sum has { %counts{$sum} / 2 } fourth power sums";
}

This works, and is quite fast in Niecza. (It's much slower in Rakudo for some reason, though it still finishes in about ten seconds.) I love that first line, which constructs a lazy infinite list of fourth powers, then truncates it at 635318657. Then it uses X+ to build all the sums of two fourth powers less than (or equal to) 635318657. The rest of the code is pretty mundane, I fear.

I decided to try to use classify to get rid of the explicit loops, resulting in this code:

my @fourth-powers = (1..*).map(* ** 4) ...^ * > 635318657;
my @results = (@fourth-powers X+ @fourth-powers).classify({ $_ }).pairs.grep(*.value > 2);
say @results.map({ $_.key ~ " " ~ $_.value.Int / 2 ~ " times" });

Unfortunately, this doesn't work in Niecza. So now I've got some bugs to fix...

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One Response to “Sums of Fourth Powers”

  1. colomon Says:

    Fixed classify in Niecza, and it turns out the .pairs in line 2 (second version) is unnecessary.

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