studies on largest prime factor ?

[kurtulmehtap]

Dear All,
I am looking for papers on the largest prime factor of a mersenne number.
As far as I know the largest prime factor has to be less than the square root of the mersenne number.
Are there any tighter bounds?

Thanx in advance

[ET_]

Originally Posted by kurtulmehtap

Dear All,
I am looking for papers on the largest prime factor of a mersenne number.
As far as I know the largest prime factor has to be less than the square root of the mersenne number.
Are there any tighter bounds?

Thanx in advance

IIRC, the bound applies to the largest penultimate prime factor of a Mersenne number.

2^11-1 = 2047 = 23 x 89 and 89 > sqrt(2^11 - 1)

Otherwise, the upper limit of trial-factoring for a number is sqrt(n).

Mersenne factors have the property of being of the form 2kp+1 (k=1,2,3... and p= the exponent of the number), so the upper limit may as well be lowered to 2kp+1 < sqrt(p) (where we have 2kp+1 < sqrt(p) < 2(k+1)p+1)

There are other heuristics, like the factor should be 1 or 7 mod 8, or [16 different values] mod 120, or [more different values] mod 420 (or 4620). Those values for the factor automatically rule out multiples of 3, 5, 7, 11.

Luigi

[science_man_88]

Originally Posted by kurtulmehtap

Dear All,
I am looking for papers on the largest prime factor of a mersenne number.
As far as I know the largest prime factor has to be less than the square root of the mersenne number.
Are there any tighter bounds?

Thanx in advance

not completely accurate the largest can be more but to not be prime it must have one below the sqrt. all factors of prime exponents mersennes must be of the form 2kp+1 and also being 8x1 and 6y1 ( and yes I may be missing something).

[science_man_88]

http://www.google.ca/search?sourceid...rsenne+numbers is what I did and the last one on my first page of results seems good.

[R.D. Silverman]

Originally Posted by ET_

IIRC, the bound applies to the largest penultimate prime factor of a Mersenne number.

There is nothing special about Mersenne numbers in this regard.
Why do people seem to think that they are special? NO
number (trivially) can have two prime factors greater than its square root.

There are other heuristics, like the factor should be 1 or 7 mod 8, or [16 different values] mod 120, or [more different values] mod 420 (or 4620). Those values for the factor automatically rule out multiples of 3, 5, 7, 11.

Luigi

These restrictions are NOT 'heuristics'. They are THEOREMS.

[ET_]

Originally Posted by R.D. Silverman

There is nothing special about Mersenne numbers in this regard.
Why do people seem to think that they are special? NO
number (trivially) can have two prime factors greater than its square root.




These restrictions are NOT 'heuristics'. They are THEOREMS.

Sorry for the confusion...

In the first answer I specified Mersenne numbers because that was the question. You are right when you say that the same apply to all numbers.

And about my use of the term heuristics... I used that term when I first approached the problem, many years ago. Now I should have learnt that "Theorem" is the right word: thank you for your making it even clearer to everybody.

Finally, I am glad that your answer dealt about terms use, and not about trivial math errors in my answer.

Luigi

[R.D. Silverman]

Originally Posted by ET_

IIRC, the bound applies to the largest penultimate prime factor of a Mersenne number.

2^11-1 = 2047 = 23 x 89 and 89 > sqrt(2^11 - 1)

Otherwise, the upper limit of trial-factoring for a number is sqrt(n).

Mersenne factors have the property of being of the form 2kp+1 (k=1,2,3... and p= the exponent of the number), so the upper limit may as well be lowered to 2kp+1 < sqrt(p) (where we have 2kp+1 < sqrt(p) < 2(k+1)p+1)

The inequality posted just above is total nonsense. May I suggest that
you proofread your posts before submitting them?

[R.D. Silverman]

Originally Posted by kurtulmehtap

Dear All,
I am looking for papers on the largest prime factor of a mersenne number.
As far as I know the largest prime factor has to be less than the square root of the mersenne number.
Are there any tighter bounds?

Thanx in advance

May I suggest that in the future you should CHECK your
claims before posting? 15 seconds of checking factorizations of
2^n-1 (from published tables) would have revealed that your claim
about the largest prime factor being less than the square root was
nonsense. Finding counter-examples is trivial.