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March 27, 2017

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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 - ,

    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 - ,

    111 is incorrect

  • Maths - ,

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

  • Maths - ,

    sorry, the answer is 1 :P

  • Maths - ,

    thanxxxx

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