Odd Prime Triples: Find & Explore Solutions!

In summary, the conversation revolves around finding triples of odd primes that satisfy certain conditions involving divisibility. Two such triples are given and it is stated that there are no other triples with the first prime being less than 10^7. The question is posed if there are any other triples and if there is more that can be said about them. The conversation also mentions the quadratic residue theory and an observation that 2, 5, and 13 also work, but 2 is not an odd prime. Finally, the conversation briefly mentions the BEAL CONJECTURE and a previous attempt at solving it.
  • #1
steiner1745
1
0
I got this question from another
forum and it's driving me crazy.
Find all triples of odd primes,
p,q,r such that
p^2+1 is divisible by q, q^2+1 is divisible by r
and r^2+1 is divisible by p.
Two such triples are 5,13,17
and 17,29,421. If we assume
p<q<r, then there are no other
such triples with p<10^7.
Are there any others?
Anyone have any ideas?
From quadratic residue theory
we know that p,q,r are all
congruent to 1(mod 4).
Can we say more?
 
Last edited:
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  • #2
(2,5,13) also works...
 
  • #3
micromass said:
(2,5,13) also works...

nice observation but 2 is not an odd prime.

Years ago i thought i solved the BEAL CONJECTURE because i found 3^5 + 10^2 = 7^3

Then my math prof. pointed out ALL exponents must be integers greater than 2.:smile:
 

1. What are odd prime triples?

Odd prime triples are a set of three odd prime numbers that are in the form of (p, p+2, p+4) where p is a prime number. These triples are also known as prime constellations or prime triplets.

2. How do you find odd prime triples?

To find odd prime triples, we need to first identify a prime number. Then, we add 2 and 4 to that prime number to form a triple. This triple is only considered an odd prime triple if all three numbers in the set are prime.

3. Are there any restrictions on the values of odd prime triples?

Yes, there are two main restrictions on odd prime triples. First, all three numbers in the triple must be odd primes. Second, the difference between each number in the triple must be exactly 2. This means that the only possible values for the first number in the set are primes that are congruent to 1 or 3 modulo 6.

4. How many odd prime triples are there?

The number of odd prime triples is infinite. However, as we go higher in the range of numbers, the density of the triples decreases. This means that there are fewer and fewer odd prime triples as we increase the range of numbers.

5. What is the significance of odd prime triples in mathematics?

Odd prime triples have been studied extensively by mathematicians as they are a special case of prime constellations. These triples have also been used in number theory and cryptography. They also have implications in Goldbach's conjecture, a famous unsolved problem in mathematics.

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