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TASK #1 - Eban Numbers
Write a script to generate all Eban Numbers <= 100.
An Eban number is a number that has no letter 'e' in it when the
number is spelled in English (American or British).
Example
2, 4, 6, 30, 32 are the first 5 Eban numbers.
MY NOTES: Very easy, no doubt there are CPAN modules to "speak" numbers
in English, but let's do it from first principles..
TASK #2 - Cardano Triplets
Write a script to generate first 5 Cardano Triplets.
A triplet of positive integers (a,b,c) is called a Cardano Triplet if
it satisfies the below condition.
cuberoot(a+b.sqrt(c)) + cuberoot(a-b.sqrt(c)) = 1
Example
(2,1,5) is the first Cardano Triplet.
MY NOTES: Ok, two mildly tricky things:
1. real arithmetic means "=" is imprecise, we can't even use rationals as
sqrt() and cuberoot() are involved, so we'll need our old abs(diff)<epsilon
trick.. and
2. the question says (2,1,5) is the "first" Cardano triplet - in what order?
The answer to the latter question sets the basic structure of the program.
Perhaps we should "order by minimum sum of triplet numbers"? ch-2.pl does
that, effectively generating all (a,b,c) where a+b+c=SUM for gradually
increasing values of SUM, testing whether each (a,b,c) triple is a Cardano
triple. It works perfectly well - but quite slowly.
I then built a SECOND VERSION (ch-2FAST.pl) which uses a much more efficient
parameterised version of Cardano Triplets that I found on the Internet.
See the top comments in that for a longer explanation, but it turns out
that we can (nearly) directly pick out Cardano triples by calculating:
a=3k+2 and x=(k+1)**2(8k+5) for each k
(where x represents bsquared*c)
then we need to break x down into b and c - noting that several (b,c) pairs
may result from a single value of x.
How much faster is this than ch-2.pl? For n=40, ch-2 takes 30 seconds where
ch-2FAST takes just under 2 seconds! And this gets better as n increases,
ch-2FAST takes 6.9s for n=100, but I gave up on running ch-2 40 after a couple
of minutes when it had only found about 60 triples.
Can anyone find a faster version?
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