106k views
1 vote
Prove or disprove: n²+41n+41 is a prime number for every integer n.

1 Answer

4 votes

Final answer:

The claim that n²+41n+41 is a prime number for every integer n is disproven by substituting n = 41 into the expression, resulting in a product that is clearly not prime, as it is 41 multiplied by another integer.

Step-by-step explanation:

In order to disprove the claim that n²+41n+41 is a prime number for every integer n, we can find a counterexample. Consider when n is equal to 41. Substituting n = 41 into the expression gives us 41² + 41(41) + 41, which simplifies to 41(41 + 41 + 1). Since the product of 41 and 123 is clearly not a prime number (it is 41 multiplied by another integer), this shows that the expression is not prime for every integer n. Therefore, we have disproven the claim by finding a specific integer n for which the expression is not prime. We can generalize this approach and state that for any integer n that is also a prime number, and n > 1, the expression will not yield a prime number since it will be divisible by n.

User Satendra
by
8.0k points

Related questions

asked Nov 7, 2024 113k views
Idealmind asked Nov 7, 2024
by Idealmind
8.3k points
2 answers
2 votes
113k views
asked Sep 4, 2016 43.8k views
Xraynaud asked Sep 4, 2016
by Xraynaud
7.5k points
2 answers
3 votes
43.8k views