Factors bounds

[jtavares]

At some time ago i conducted a study for finding smaller bounds on the prime factors for integer composite numbers of type n=p*q, n>0

It is already known that 1<=p<=sqrt(n) and sqrt(n)<=q<=n.

I was able to find smaller bounds by an empirical method (still remains to conduct more tests to validate).

My question is: have these results pratical interest?
or
have these results cientific interest that justifies a publication in a jounal?

[R.D. Silverman]

Originally Posted by jtavares

At some time ago i conducted a study for finding smaller bounds on the prime factors for integer composite numbers of type n=p*q, n>0

It is already known that 1<=p<=sqrt(n) and sqrt(n)<=q<=n.

I was able to find smaller bounds by an empirical method (still remains to conduct more tests to validate).

My question is: have these results pratical interest?
or
have these results cientific interest that justifies a publication in a jounal?

(1) You gave, above, the only bounds that are POSSIBLE to give as a
function of n.

(2) Your claim of "smaller bounds", is therefore nonsense. Furthermore, one
does not validate mathematics by "conducting tests".

(3) You ask whether "these results" have any practical interest, but you
have not presented any results.

[jtavares]

Originally Posted by R.D. Silverman
(1) You gave, above, the only bounds that are POSSIBLE to give as a
function of n.

(2) Your claim of "smaller bounds", is therefore nonsense. Furthermore, one
does not validate mathematics by "conducting tests".

(3) You ask whether "these results" have any practical interest, but you
have not presented any results.

(1) Indeed, but mathematics is still evolving!

(2) There were many unproven and usefull theories until... someone proved it!

(3) But is my intention to do it. I shall apply it to the infamous RSA challeng numbers, both factored and yet to be factored.

I just want some feedback for the importance of establishing new bounds on factors, with or without a well founded mathematic theory behind.

[xilman]

Originally Posted by jtavares

(1) Indeed, but mathematics is still evolving!

(2) There were many unproven and usefull theories until... someone proved it!

(3) But is my intention to do it. I shall apply it to the infamous RSA challeng numbers, both factored and yet to be factored.

I just want some feedback for the importance of establishing new bounds on factors, with or without a well founded mathematic theory behind.

Ok. Here is some feedback. Bob Silverman is completely correct in his claim and you are entirely wrong in your claim.

It is possible to prove, rigorously, that the known bounds are the best possible.

An example, which you should be able to understand for yourself: let an integer N be the square of a prime p. That is, N=p*p. The existence of such N, and I have given you an explicit example, shows that if N=p*q where p>=q, then best possible upper bound for the smaller factor of N is sqrt(N) and the best possible lower bound on q is also sqrt(N)

If, by some chance, you do not understand symbolic algebra and, therefore, the example given above, please explain how the smallest prime factor of 121 is less than sqrt(121). Remember that 1 is not prime and is a factor of every integer.

Paul

[jinydu]

xilman is right. But if n is a composite of some particular form (ex. (2^p) - 1 ), you may be able to set better bounds.

[cheesehead]

Originally Posted by jinydu

... if n is a composite of some particular form (ex. (2^p) - 1 ), you may be able to set better bounds.

Not when the composite is the product of two primes p and q, as specified above. Then, as Silverman said, there are no better bounds.

If there are more than two prime factors, then you have a different case. But the original posting specified "n=p*q" (with the implication by context that p and q were both prime).