How Many Triplets?

Algebra Level 3

How many triplets of prime numbers exist such that the sum of their squares is $2019?$

In other words, how many ways can we write $2019$ as the sum of the squares of three primes?

3 5 6 2 1 4

1 solution

Hana Wehbi
Jan 3, 2019

Note that:

$7^2+11^2+43^2=2019\\ 7^2+17^2+41^2=2019\\ 13^2+13^2+41^2=2019\\11^2+23^2+37^2=2019\\17^2+19^2+37^2=2019\\23^2+23^2+31^2=2019$

Thus, $2019$ can be written in 6 ways as the sum of the squares of three primes.

How do you know that 2019 is indeed the smallest positive integer that can be expressed as the sum of squares of 3 primes?

Pi Han Goh - 2 years, 5 months ago

Yeah, you are right, we need to prove it and that is another solution. I will edit it.

Hana Wehbi - 2 years, 5 months ago

I presume you mean the smallest that can be so expressed in 6 different ways? Since 12 can be expressed as the sum of squares of 3 primes for example.

Or without exhausting all the possibilities (by computer as I did), how do we know there are only 6?

Steven Perkins - 2 years, 5 months ago

Yes, we can actually write a computer program and generate all the possibilities. I just solved it through brute force technique and I was only able to come up with 6 ways.

Hana Wehbi - 2 years, 5 months ago

How do we know that there are only 6? Can it be proved that there are only 6?

Sumant Chopde - 2 years, 5 months ago

Yes, we can actually write a computer program and generate all the possibilities. I just solved it through brute force technique and I was only able to come up with 6 ways.

Hana Wehbi - 2 years, 5 months ago

How Many Triplets?

1 solution

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