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Consider Prime Ratios, Get Integers (Posted on 2023-02-04)

Find all possible triplet(s) (p, q, r) of prime numbers such that, each of:

(p² + 2q)/(q + r), (q² + 9r)/(r + p), and
(r² + 3p)/(p + q)

is a positive integer.

Prove that no further triplet is possible in conformity with the given conditions.

Note: Adapted from a problem appearing in a shortlist of Junior Balkan Mathematical Olympiad.

*** Computer program assisted solutions are welcome, but a semi-analytic methodology, that is: hand calculator and p&p, is preferred.

Submitted by K Sengupta

Rating: 5.0000 (1 votes)

Solution: (Hide)

(p,q,r)=(2,3,7) is the only possible triplet of prime numbers that satisfies all the given conditions.
For an explanation, refer to the analytic solution submitted by Brian Smith in this location.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
re: Analytic Solution	xdog	2023-02-04 11:58:45
Analytic Solution	Brian Smith	2023-02-04 10:53:13
computer findings	Charlie	2023-02-04 09:54:58
An analytical start	Steve Herman	2023-02-04 09:44:37

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