Does what hold true? On one hand you have Dirichlet’s theorem (on infinitely many primes in arithmetic progressions); on the other hand there is the more obvious fact that if p is prime and p divides a bigger number then that number is not prime. Or, one way to interpret your question is whether the product of primes +/- 1 is again prime, which you yourself observed is not true.
The product of two primes +/- 1 cannot be prime, unless at least one of those primes is 2. But, the product of two primes +/-2 can be prime, and the few examples that I have worked out have worked out that way. 3x5 +/- 2 = 13 and 17, 3x7 +/-2=19 and 23. 5x7+/-2=33(not a prime) and 37. I was just wondering if that sequence had any significance. Dirichlet’s theorem is interesting, but not what I was actually talking about, I don’t think.
Every (sufficiently-large) prime must be equal to 1, mod 2.
Every (sufficiently-large) prime must be equal to 1 or 5, mod 6.
Every (sufficiently-large) prime must be equal to 1, 7, 11, 13, 17, 19, 23, or 29, mod 30.
I could go on with the lists for 210, 2310, and so on, but it gets cumbersome.
Many of the classic problems in number theory are full of “except for” conditions.
E.g., Fermat’s last theorem: There are no positive integers a, b, c, n such that a[sup]n[/sup] + b[sup]n[/sup] = c[sup]n[/sup] for n > 2. Lots of examples for n = 1 or 2. (Or allowing a, b, or c be 0.)
Goldbach’s conjecture: All even integers greater than 2 can be expressed as the sum of two primes.
Interesting, I seemed to see a pattern, I guess others have as well. There wouldn’t happen to be a Numberphile or similar type video about this, would there?
It’s not really any more interesting than the Sieve of Erastothenes. Less so, in fact, since the Sieve is a practical implementation of the same idea.
On the topic of small exceptional numbers, it is conjectured that 7,373,170,279,850 is the largest number that cannot be represented as the sum of the cubes of four natural numbers.
For those who want to work with the Sieve, here is an implementation in the Fractran programming language:
17/91, 78/85, 19/51, 23/38, 29/33, 77/29, 95/23, 77/19, 1/17, 11/13, 13/11, 15/14, 15/2, 55/1.
Enter 2 in the accumulator and the program will visit 2[sup]2[/sup], 2[sup]3[/sup], 2[sup]5[/sup], 2[sup]7[/sup], 2[sup]11[/sup], 2[sup]13[/sup], 2[sup]17[/sup], etc. in order.
This might be a good place to point out that every number is a small number.
Oh yeah? What about 7?
Also, consider numbers that are uncomputable; the standard textbook example is one of Radó’s functions, or you could use Rayo’s function, etc. Now consider a number like Rayo(10^100). It is finite, but one could argue that it legitimately qualifies as “not small”. It’s not like you can compute it or even say quite how big it is.
But, you could confidently say that there are more numbers that are bigger than it than are smaller than it, by a ratio of infinity over 1.
So, Rayo(10^100), while being an aneurysm causing big number, is still a pretty tiny number, compared to what else is out there.
Certainly, if all finite numbers are small, then every natural number is small, because every natural number is finite, which is what Chronos said. How about this, though: consider a model of non-standard arithmetic. In this model, there exists a number x such that x > n for every standard number n. We might call the number x a “non-small number”, although there are plenty of bigger numbers, like x+1, x+1000, x+x+x, or whatever.
This voids the meaning of small. One could just as well say that all animals are small, all stars are small, etc. In other words, “small” has no use whatsoever.
Except Graham’s Number. Anything smaller is small, and anything larger is close enough to infinity to be infinity, which simply goes on forever, and thus is easy to imagine.
No, not all animals or stars are small. I can say, for instance, that there are a lot more stars smaller than the Sun than there are stars larger than the Sun, and that, therefore, the Sun is a large star. I can say that there are a lot more animals smaller than me than animals bigger than me, and that I am therefore a small animal. But there are a lot more numbers larger than Graham’s number than numbers smaller than it, and therefore, Graham’s number is a small number.