Generating novel yet concise sequences with + and x

[Loren Booda]

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.

 

[dodo]

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.

 

[Loren Booda]

Whoa, Dodo! Thank you for your insightful analysis.

It makes me want to create more.