Problem #PRU-100577

Problems Number Theory Divisibility Divisibility of a number. General properties Division with remainders. Arithmetic of remainders Euler's theorem

Problem

Look at this formula found by Euler: $n^{2} + n + 41$ . It has a remarkable property: for every integer number from $1$ to $21$ it always produces prime numbers. For example, for $n = 3$ it is $53$ , a prime. For $n = 20$ it is $461$ , also a prime, and for $n = 21$ it is $503$ , prime as well. Could it be that this formula produces a prime number for any natural $n$ ?

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