It is well known that there are infinitly many primes as shown by many theorem and proofs in Mathematics, Really prime number play an important role in differents area of mathematics particulary number theory and also it gives huge contributions in the side of it's application , Now my question may is trivial , I want to know What we can get as results in mathematics if we have finit primes ? Does Results being better ?
Salam Rafik,
Good question,
A finite set of primes means two things:
Either there are finite set of natural numbers and we have yet to find this hypothetic superior bound which might be extremely large but finite and then infinity is just an illusion. Or that the set of natural numbers is indeed infinite and then there are an infinity of them unfactorisable by primes.
If primes are finite, this means the following:
There exists a finite number of primes P1, P2, ...,Pk such that for every number n belongs to N, n can be expressed uniquely as
n = P1^{a₁}P2^{a₂}....Pk^{a_{k}, where a_{i} ≥ 0.
This wrong assumption, or wrong axiom.
If we add a wrong axiom to our mathematical system, it becomes inconsistent, and almost all related facts become wrong.
HOW?
Assume that Primes are finite say P1, P2, ...,Pk,
m = P1 P2 ...Pk+1 does not belong to this new system of numbers!!!
m is not divisible by any one of them!!
Our new mathematics will not work anymore.
Imagine the terrible consequences that will happen; it will delete and destroy the accumulative knowledge for humanity in all branches of science for along long period.
Best regards
Dear Peter,
In general
Units are the invertible elements in a ring R.
Here we have Z/nZ the ring of integers modulo n.
The set of all integers coprime to n are called units and form a multiplicative group denoted by (Z/nZ)x, the order of this group is ϕ(n) the Euler totient function.
Example: (Z/12Z)x
The set of units are (Z/12Z)x = {1,5,7,11} a multiplicative group.
The number of units is ϕ(12)=4.
inverse of 1 is 1 because 1x1=1mod(12)
inverse of 5 is 5 because 5x5=1mod(12)
inverse of 7 is 7 because 7x7=1mod(12)
inverse of 11 is because 11x11=1mod(12)
In Zn:
We have the following useful lemma
[a]n | [b]n if and only if gcd(a,n) | gcd(b,n).
And
We have the following useful theorem:
The set of irreducible elements of Zn are
{ [a]n}∈ Zn,there exists a prime divisor p of n such that
p²|n and gcd(p,n)=1}
And
The set of prime elements of Zn are
{ [a]n∈ Zn,there exists a prime divisor p of n such, p
In the group of integers with only addition there's only one prime, the number 1. It's not generally thought of as a prime but in this context, if we generalise the notion of a prime number then having a single prime number 1 generates the integers with addition and subtraction.
The primes when thought of among the rational numbers excluding zero: Q∖{0} with multiplication, are the generating elements of a multiplicative group - an algebraic object in which every element has an inverse. This is the free abelian group generated by infinitely many primes.
If you wanted to understand better what it would mean if there were finitely many primes, the most sensible answer would be to continue the analogy to the free abelian group generated by some finite subset of the primes, say choose {2,3,5}. Then you would have a number system in which you could generate any number of the form:
2x⋅3y⋅5z2x⋅3y⋅5z where x,y,z∈Z. This would give you a proper subset of Q∖{0}Q∖{0} which could contain any of the numbers in the following set of examples:
{2,12,3,127,52,150,1150…}
Your instinct is in fact correct, there are many circumstances at an advanced level where mathematicians do consider there to only be finitely many primes - at least within some number system they are considering, and so doing can make certain tasks easier. An example might be if you can demonstrate that a solution to some problem is only drawn from some finite subset of the primes, then you might choose an algebraic object only containing finitely many.
@ Rafik
[ In the group of integers with only addition there's only one prime, the number 11. It's not generally thought of as a prime but in this context, if we generalise the notion of a prime number then having a single prime number 11 generates the integers with addition and subtraction].
This is statement is not true.
What do you mean by 11 is the only prime? In which group?
My previous answer was posted to Peter following his example
about Z12 . He asked for any comment.
P→Q means ¬P∨Q . This means that if there are finitely many primes(P), ¬P will be true and therefore any Q will satisfy P→Q. Therefore, everything will be true.
@Isamm kaddoura , Just i did a wrong typo , I meant the number 1 not 11, all my mathematical expressions were appeared multiplied , I have edited it and it's seems all things are fixed , Thanks for ur attention
@ Peter,
(*)That a|b mod n iff (a,n)|(b,n) is a wonderful lemma. Where's it from, or did you invent it?
No, I am not the inventor. It is a consequence of the definition.
Almost all mentioned theorems, lemmas, etc. in my previous answer are well- known.
Best regards
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