4

Part of 2022 Austrian MO National Competition

Problems(2)

4q - 1 is prime and (p + q)/(p + r)= r - p, diophantine in primes

Source: 2022 Austrian Federal Competition For Advanced Students, Part 1 p4

Find all triples

(p, q, r)

of prime numbers for which

4q - 1

is a prime number and

\frac{p + q}{p + r} = r - p

holds.
(Walther Janous)

prime numbersnumber theoryDiophantine equationdiophantine

infinite many primes divide P(n)

Source: 2022 Austrian Federal Competition For Advanced Students, Part 2 p4

Decide whether for every polynomial

P

of degree at least

1

, there exist infinitely many primes that divide

P(n)

for at least one positive integer

n

.
(Walther Janous)

number theoryInteger Polynomialpolynomial