The four right-angled triangles with sides $(9,12,15)$, $(12,16,20)$, $(5,12,13)$ and $(12,35,37)$ all have one of the shorter sides (catheti) equal to $12$. It can be shown that no other integer sided right-angled triangle exists with one of the catheti equal to $12$.
Find the smallest integer that can be the length of a cathetus of exactly $47547$ different integer sided right-angled triangles.
This question involves a very important concept from number theory – the distribution of numbers with a lot of divisors.
Primitive Pythagorean Triplelets (PPTs) have the property a^2 + b^2 = c^2, which only holds for integer values of a, b, and c where “a” and “b” are coprime (they share no common factor except 1) and “c” is a odd number.
It’s possible to express the a, b, and c in terms of p and q where a=2pq, b=p^2-q^2, and c=p^2+q^2, where p and q are coprime and of different parity (one is even, the other is odd).
Now let one leg (cathetus) of the triangle be “n” which equals 2pq.
To solve this question, we need to determine all the possible ways of factorizing n into p and q satisfying the conditions p>q, (p,q)=1 and p and q are of different parity.
For example leg of length 12 belongs to 4 Pythagorean triplets because it can be factorized in four ways: (1,12), (2,6), (3,4) satisfying the conditions.
If we want to have 47547 triplets for a shortest possible leg’s length then we would want to find the smallest number with 47547*2 factors (we double because we are taking both p>q and pMore Answers:
Hollow Square Laminae I
Hollow Square Laminae II
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