+4622 How many integers n are there such that the quantity \(\lvert 2n^2 + 23n + 11 \rvert\) is prime?
+129899 Not totally sure about this...but.....here's my best attempt...
abs [ 2n^2 + 23n + 11 ] factor
abs(2n + 1) * abs(n + 11)
Note that this will be a possible prime if either factor = ±1
But 2n + 1 will = 1 only when n = 0.....and the other factor will = abs ( 11 ) = 11
So....when n = 0, the result will be prime, i.e, 11
And 2n + 1 will equal - 1 when n = -1.....and the other factor will = abs(-1 + 11) = 10......but this isn't prime
And n + 11 will equal 1 when n = -10
And the other factor will be 2(-10) + 1 = -19 which is prime for abs (2n + 1) = abs (2*-10 + 1) =
abs(-19) = 19
And n + 11 will = - 1 when n = -12 ....so abs (-12 + 11) = abs(-1) = 1
And the other factor will be abs (2(-12) + 1) = abs (-23) = 23
So.....this will be prime when n = -12
So.....the integers producing prime results for abs [ 2n^2 + 23n + 11 ] are
n = 0 , n = -10 and n = -12
EDITED ANSWER.....still don't know if it's correct, or not....!!!!!!
+199 We first note that \(2n^2 + 23n + 11\) factors as \((2n + 1)(n+ 11)\) . (We can find these factors using the rational root theorem.) Thus we have \( \lvert 2n^2 + 23n + 11 \rvert = \lvert 2n + 1 \rvert \cdot \lvert n + 11 \rvert . \)Now, each of the factors on the right hand side of this equation is an integer. It follows that the left hand side is a prime number if and only if one of the right hand factors is 1 and the other one is a prime number. Thus we must either have \(2n + 1 = \pm 1\) , or \(n +11 = \pm 1\). We consider these cases separately.
If 2n+1, then n=0, and n+11=11, which is prime. Thus this value of n works.
If 2n+1=-1 , then n=-1, and n+11=10 , which is not prime. Therefore we have no solution in this case.
If n+11=1 , then n=-10 , so 2n+1=-19 . Since 19 is prime, we obtain a valid solution in this case.
Finally, if n+11=-1, then n=-12 , and 2n+1=-23. Since 23 is prime, this value of n works.
Thus there are exactly \(\boxed{3}\) values of \(n\) that work: 0, \(-10\), and \(-12\) ; and these give the prime numbers 11, 19, and 23.
