Show that if x is an integer then x2+x-41= 0 produce prime numbers.

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By mathematical induction

X2 +X-41=0

Let X=n

P(n)=n2+n-41

P(1)= 2-41=-39 it's not a prime number this statement is not true for n=1

P(2)= 6-41= -35 not prime number this statement is not true for n=2

P(3)= 12-41=-29 prime number. This statement is true for n=3

P(4) =20-41=-21 not prime number, this statement is not true for n=4

P(5)= 30-41=-11 prime number, this statement is true for n=5

n ∈ k

n∈ 2k +1

P(2k+1) = (2k+1)2 +(2k+1)-41 which is true for n∈ 2k+1

Therefore X2 +X-41 is true for n∈ 2k+1