Generating novel yet concise sequences with + and x

• Loren Booda
In summary, the conversation discussed a pattern of adding and multiplying two numbers and how it can lead to infinitely many prime numbers. This pattern can be generated by using coprime numbers as the initial values, and each new left-hand number will exhibit a new prime number in its factorization. The conversation also mentioned the simplicity of the algorithm used to create these sequences and how it can lead to the creation of more patterns.
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.

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.

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Whoa, Dodo! Thank you for your insightful analysis.

It makes me want to create more.

1. How does using "+" and "x" help in generating novel sequences?

Using "+" and "x" allows for the combination and repetition of existing elements in a sequence, creating new and unique sequences that may not have been previously thought of.

2. Can this method be applied to any type of sequence, or only specific ones?

This method can be applied to any type of sequence, as long as the elements can be combined and repeated using the "+" and "x" symbols.

3. Are there any limitations to using "+" and "x" in sequence generation?

One limitation is that the resulting sequences may not always make logical sense or have a clear meaning. Additionally, the length of the sequence may be limited by the number of elements available for combination and repetition.

4. How can this method be used in scientific research?

This method can be used in various scientific fields, such as bioinformatics, to generate new sequences for studying genetic patterns or in chemistry for creating new molecular structures. It can also be used in computer science for creating algorithms or in linguistics for generating new language patterns.

5. Are there any other methods besides using "+" and "x" for generating novel sequences?

Yes, there are other methods such as randomization, substitution, and mutation that can also be used for generating novel sequences. Each method may have its own advantages and limitations, and the choice of method may depend on the specific goals and needs of the research.

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