71–80 of 139 posts
They've included 1 as not a prime. That was a fairly recent decision.
One is not a prime number. If you allow one to be a prime number, then you can no longer say that each natural number has a unique prime factorization. This makes the concept of prime numbers much more useful when one is excluded.
Endlessly saying "primes other than 1" gets kind of tedious.
Here is a "theorem" I learned: every number up to 100 which looks prime, is prime, except 91. Does anyone recall its name?
Earlier quoted context omitted.
One is not a prime number. If you allow one to be a prime number, then you can no longer say that each natural number has a unique prime factorization. This makes the concept of prime numbers much more useful when one is excluded.
> If you allow one to be a prime number, then you can no longer say that each natural number has a unique prime factorization. I don't dispute that, by definition, 1 is not prime, but I don't see how this statement would follow if we considered it prime. Edit: it seems more like it would be that every factorization would implicitly have 1^n tacked onto it, and while that isn't exactly useful, it doesn't break the gam…
Is there a reason we're obsessed with primes beyond aesthetics? Why does this set of numbers garner all the headlines as opposed to some other arbitrary integer sequence like the Recamán numbers [0] ? If tomorrow someone discovered a closed-form equation for the nth prime, how would mathematics/the world change? [0] https://en.wikipedia.org/wiki/Recamán%27s_sequence
In algebra, the integers mod p are a finite field (addition, subtraction, multiplication, and division are defined) if and only if p is prime.
Primality in algebra also exists in a more general form with prime ideals. An ideal is the set of elements of a commutative ring in which any element of the ideal multiplied by any element of the ring is still in the ideal; there is a sort of 'closed' property. Even numbers form an ideal because if you multiply any number by an even number, you get an even number. For a prime ideal, if ab is in the ideal, a is in the ideal or b is (similar to how if a prime number divides ab, it divides a or b).
They have a number of interesting properties. For example, for a ring homomorphism (a function from one ring to another that preserves relationships between elements of the two rings), the pre-image of a prime ideal is also a prime ideal.
Earlier quoted context omitted.
> If you allow one to be a prime number, then you can no longer say that each natural number has a unique prime factorization. I don't dispute that, by definition, 1 is not prime, but I don't see how this statement would follow if we considered it prime. Edit: it seems more like it would be that every factorization would implicitly have 1^n tacked onto it, and while that isn't exactly useful, it doesn't break the gam…
That breaks the uniqueness of the prime factorization. Mathematicians love uniqueness, almost as much as existence.
78. Who did better?
1 is non-prime by definition.
You correctly sorted 17 numbers.