Posted by ianian on Monday, April 29, 2013 at 4:13am.
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?
- Maths - Reiny, Monday, April 29, 2013 at 8:26am
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
- Maths - tsong, Monday, April 29, 2013 at 11:13am
111 is incorrect
- Maths - PD, Tuesday, April 30, 2013 at 2:57pm
@Tsong, the answer is not 111, the answer is 3 :P
- Maths - PD, Tuesday, April 30, 2013 at 2:58pm
sorry, the answer is 1 :P
- Maths - abc, Saturday, May 4, 2013 at 2:57am
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