Question

Consider all 3-term geometric sequences with first term 1 and with common ratio the square of an integer between 1 and 1000. How many of these 1000 geometric sequences have the property that the sum of the 3 terms is prime?

Answer 1

so we are looking for
1 + x^2 + x^4 being a prime


x^4 + x^2 + 1
= x^4 + 2x^2 + 1 - x^2
= (x^2 + 1)^ - x^2 ---- a difference of squares
= (x^2 + 1 +x)(x^2 + 1 -x)

if x = 1 , we get
(1+1+1)(1) = 3 --- a prime number

for any other value of x, the value of each bracket > 1
and 1+x^2 + x^4 is the product of at least two factors

thus : 1 + 1^2 + 1^4 is the only such case
1 1 1 is the only case

Answer 2

111 is incorrect

Answer 3

@Tsong, the answer is not 111, the answer is 3 :P

Answer 4

sorry, the answer is 1 :P

Answer 5

thanxxxx