The Lucas-Lehmer Prime Number Test
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MathematicsNumber TheoryPrime Numbers
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Prime Numbers
The article from Scientific American explains methods to identify prime numbers without using a computer, and the HN community shows interest in the topic with a positive score.
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That is it. That is all. Pish posh.
If a number is not prime, then it is the product of at least two numbers smaller than itself.
If any of them are larger than its square root, all others must be smaller, or their product would be larger than the candidate prime.
Ergo, just check that the candidate is not evenly divisible by any number equal or lower than its square root.
This reasoning holds, independent of scale.
QED. Check mate. Shazam.
If you have the time.
Or if we can expand quantum superposition algorithms from 2^N states, for quantum circuits with N control qubits, to 2^(T*N) superpositions over T time steps, via some kind of superposition tree recursion. The number of superpositions increasing exponentially for T steps (and then reducing for another T steps) on a single recursive physical circuit.
That is not supported by the physical laws we have, but it is an interesting idea.
What does this part mean? For example 57.
The standard divisibility rule for 3, 6 and 9 in base 10 is to sum the digits until you only have one left and check if it's one of those. Here, 5+7=12, 1+2=3, so 57 is divisible by 3.
Math is crazy!... still don't want to study it though!
My fun fact is that this type of operation (repeatedly applying a child operation until you reach a fixed point) is called persistence.
https://en.wikipedia.org/wiki/Persistence_of_a_number
The fixed point here being that if you add up a list of 1 digits, you'll always reach the same number (`sum([1]) = 1`). The best known is probably the hailstone sequence.
https://en.wikipedia.org/wiki/Collatz_conjecture
I'm partial to multiplicative persistence.
https://www.youtube.com/watch?v=Wim9WJeDTHQ [15m]
http://calcoastrails.com/
(factorial(170,141,183,460,469,231,731,687,303,715,884,105,726) + 1)%(170,141,183,460,469,231,731,687,303,715,884,105,727) == 0
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