# Primitive integer triangles

Consider the triangles with integer sides $a$, $b$ and $c$ with $a \leq b \leq c$. An integer sided triangle $(a,b,c)$ is called primitive if $gcd(a,b,c)=1$. How many primitive integer sided triangles exist with a perimeter not exceeding $10 000 000$?

I am trying to solve this on euler project. I am wondering what is the best way to go to find the valid triples for constructing a triangle. Of course you can do nested for loops but that is not efficient. Any pointers would help.

Thanks

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## 3 Answers

This is problem 276. I haven't solved it, but the first thing to realize is you don't have to loop over c-from a and b you should be able to find the number of solutions. Of course, you don't have time to loop over both a and b, but you may be able to loop over a. I would start by doing a brute force run up to 100 or 250-you should have time for that. Keep track of the number of solutions for each a. Then try to find a formula for the number of triangles that include a. This will probably be based on the factors of a.

Hope this helps. I did a lot of them, but have gotten less energetic now.

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HINT $\rm\ gcd(a,b,c) = gcd(10^7-b-c,b,c) = gcd(10^7,b,c)\:,$ so a triple will be primitive unless $\rm\:b,c\:$ are both divisible by $2$ or $5$. The longest leg $\rm\:c\:$ must be less than half the perimeter $\rm\:P\:$ and at least $\rm\:P/3\:.\:$ Therefore, for each $\rm\:c\:$ in this range, count the values of $\rm\ b = c,\: c-1,\ldots, \lceil (P-c)/2\rceil\$ such that $\rm\:b,c\:$ have neither $2$ nor $5$ as a common divisor. It depends only on $\rm\:b,c\:$ modulo $\ldots$

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The perimeter need not be $=10^7$, it just has to be $\leq 10^7$ – Ross Millikan Jan 5 '11 at 21:35
Indeed, but once should able to use similar sieve-like ideas to avoid gcd computations. But there is probably a better way. – Bill Dubuque Jan 5 '11 at 22:06
Does it mean that the solution wont work for p <= 10^7? – Jinesh Parekh Jan 6 '11 at 6:07

There is a formula which gives you all the Pythagorean triples. That should cut down your search space a bit.

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The triangles need not be right-angled. – Chris Eagle Jan 5 '11 at 16:39