Importance of the first 3 primes

trini

hey guys, im a student engineer and have found that the laws we use are primarily based off of stats. this has led me to an interest in number theory and groupings. I was hoping someone could help me with somehing, i would like to know the relevance of the first 3 primes(2,3,5) to math in general. to be more specific, do the first three three primes play a significant role in number theory? I feel like i am being vague, so maybe i should ask, does being a multiple of 2, 3 or 5, or a power of one of these three give a number any special properties?

robert Ihnot

When it comes to checking for primes, the usual suggested calculator program says to check for 2 and 3--which is very easy, and then run a program using [tex]6X\pm1[/tex].

That way 2/3 of the integers are eliminated from further testing.

dodo

It could be worth mentioning that we have a prime number of hands (2) with a prime number of fingers on each (5) and, therefore, count everything in groups of ten.

trini

i suppose if we're taking such naturalist possibilities one could mention that our bodies are based off of the first 3 primes, in particular the number 3:

-30 bones in each arm and leg (3 x 2 x 5)
-26 vertabrae(though 2 are composites of 5 and 4, so in all there are 33 = [2³ x 3] + 3² ]
-24 ribs( 2³ x 3 )
-3 fused pelvic bones

dodo

:)
These are funny facts to know, but they lack the 'therefore' part: do they influence math in general, or number theory in particular?

trini

I don't know how these relations help math persay, but life reflects math, and given the importance of primes to number theory, it is interesting to see that primes are also found in biology, specifically human biology.

Tinyboss

Primes in general are fundamental to number theory, and naturally the first primes come up more often: half of the natural numbers are even, but not nearly so many are divisible by, say, 6547.

robert Ihnot

Tinyboss: Primes in general are fundamental to number theory, and naturally the first primes come up more often: half of the natural numbers are even, but not nearly so many are divisible by, say, 6547.

But even so there are an infinite number of integers divisible by 6547. And they could be put into 1-1 correspondance with numbers divisible by 2. [tex] 6547n\Longleftrightarrow2n[/tex]

Tinyboss

Yeah, I was playing a little fast and loose there, but what I meant was small integers, the kind like trini mentions, for instance, or the kind that show up as the order of sporadic groups, and so forth. Not that |Z/nZ| depends on n.

trini

here's my take on it, in a race to inifinity, no triplet of positive integers has more multiples than 2, 3 and 5 (not including 1 of course)

robert2734

The square of all primes greater than 5 is either (1 mod 30) or (19 mod 30).