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Prime Number - Somewhat difficult for me

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Prime Number - Somewhat difficult for me

by harsh.champ » Mon Feb 08, 2010 5:11 am
If m is an integer, how many values of m are possible for which m4 - 20m2 + 4 is a prime number?

(A)0
(B)1
(C)2
(D)3
(E)more than 3

THe OA is A.
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by shashank.ism » Mon Feb 08, 2010 5:36 am
harsh.champ wrote:If m is an integer, how many values of m are possible for which m4 - 20m2 + 4 is a prime number?

(A)0
(B)1
(C)2
(D)3
(E)more than 3

THe OA is A.
P = m4 - 20m2 + 4 = m4 - 4m2 + 4 - 16m2
= (m2 - 4)2 - (4m)2 ⇒ (m2 - 4 - 4m) (m2 - 4 + 4m)
So, P has at least two factors for any integral value of m, and also, for any value of m, any factor will not
be equal to 1.
Hence, there is no value of m for which m4 - 20m2 + 4 is a prime number. [spoiler]Answer: (A)[/spoiler]
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by ajith » Mon Feb 08, 2010 11:23 am
harsh.champ wrote:If m is an integer, how many values of m are possible for which m4 - 20m2 + 4 is a prime number?

(A)0
(B)1
(C)2
(D)3
(E)more than 3

THe OA is A.
m4 - 20m2 + 4

= (m^2-2)^2 -16m^2

= (m^2-2+4m) *(m^2-2-4m)

(now it is just a question of whether m^2+4m -2 =1 ; it is not since that equation doesnt have roots; m^2- 4m -2 =1 also doesnt have integer roots so we are safe)

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