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@ -83,7 +83,7 @@ There also exists a term **pseudoprime** -- it stands for a number which is not
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**Prime gaps**: statistically gaps between consecutive primes increase. The size of the gaps themselves make another number sequence that starts like this 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, 10, 14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 8, 6, 6, 4, 6, 8, 4, 8, 10.
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**[Fun](fun.md) with primes**: thanks to their interesting, mysterious and [random](randomness.md) nature, primes can be played around -- of course, you can examine them mathematically, which is always fun, but you can also play sort of [games](game.md) with them. For example the prime race: you make two teams of primes, one that gives 1 modulo 4, the other one that gives 3; then you go prime by prime and add points to each team depending on which one the prime falls in; the interesting thing is that team 3 is almost always in lead just by a tiny amount (this is known as Chebyshev bias, only after 2946 primes team 1 gets in the lead for a while, then at 50378 etc.). Similar thing can be done by evaluating the Mobius function: set total sum to 0, then go number by number and if it only has unique prime factors, add 1 if the number of those factors is even, otherwise subtract 1 -- see how the function behaves. Of course you can go crazy, make primes paint pictures or compose [music](music.md) -- people also like to do this with digits of numbers, e.g. those of [pi](pi.md) or [e](e.md).
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**[Fun](fun.md) with primes**: thanks to their interesting, mysterious and [random](randomness.md) nature, primes can be played around with -- of course, you can examine them mathematically, which is always fun, but you can also play sort of [games](game.md) with them. For example the prime race: you make two teams of primes, one that gives 1 modulo 4, the other one that gives 3; then you go prime by prime and add points to each team depending on which one the prime falls in; the interesting thing is that team 3 is almost always in lead just by a tiny amount (this is known as Chebyshev bias, only after 2946 primes team 1 gets in the lead for a while, then at 50378 etc.). Similar thing can be done by evaluating the Mobius function: set total sum to 0, then go number by number and if it only has unique prime factors, add 1 if the number of those factors is even, otherwise subtract 1 -- see how the function behaves. Of course you can go crazy, make primes paint pictures or compose [music](music.md) -- people also like to do this with digits of numbers, e.g. those of [pi](pi.md) or [e](e.md).
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**Can we generalize/modify the concept of prime numbers?** Yeah, sure, why not? The ways are many, we'll rather run into the issue of analysis paralysis -- choosing the interesting generalization of out of the many possible ways. Some possible generalizations include:
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