Is the number n^{2}+n+41 a prime for all positive integers

Mark Johns

Mark Johns

Answered question

2022-02-28

Is the number n2+n+41 a ' for all positive integers n?

Answer & Explanation

koffiejkl

koffiejkl

Beginner2022-03-01Added 7 answers

Step 1
In general the claim is not true.
n2+n+42=n(n+1)+41
can not be a ' number when
n(n+1)0(mod 41)
For example for n=40
n(n+1)+41
=40(41)+41
=41(40+1)
=412
is not a ' number
Similarly fo n=41
n(n+1)+41
=41(42)+41
41(42+1)
=41(43)
is not a ' number.
Pregazzix2a

Pregazzix2a

Beginner2022-03-02Added 9 answers

Step 1
f(n)=n2+n+41=n2+2n+1(n40)=(n+1)2(n40)
Just take n such that n40=k2n=40+k2
In this case:
f(n)=(n+1)2k2=(n+1k)(n+1+k)
which is composite:
k=1n=40+12=41
k=2n=40+22=44

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