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Generating novel yet concise sequences with + and x

  1. Mar 22, 2012 #1
    Choose two whole numbers, say 2 and 1.

    Add them and yield 3; multiply them and yield 2.

    Repeat using those new numbers.

    3+2=5; 3x2=6

    5+6=11; 5x6=30

    11+30=41; 11x30=330

    41+330=371; 41x330=13530 etc.

    Have such sequences been explored before? Their generation is relatively simple, with fundamental operations in an abbreviated algorithm.
     
  2. jcsd
  3. Mar 23, 2012 #2
    One interesting thing is that, if the two initial numbers are coprime, the subsequent pairs will continue to be coprime: if gcd(a,b)=1, then gcd(a+b,a) = gcd(a+b,b) = 1, and thus gcd(a+b,ab)=1.

    As the right-hand number is the product of *all* the previous numbers on the left (times the first number on the right), it follows that each new left-hand number will exhibit a new prime number in its factorization, not seen in any of the previous left- numbers... which is yet another way of proving that there are infinite primes.
     
    Last edited: Mar 23, 2012
  4. Mar 23, 2012 #3
    Whoa, Dodo! Thank you for your insightful analysis.

    It makes me want to create more.
     
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