Re: [PrimeNumbers] Re: Algebraic factoring
- djbroadhurst wrote:
> --- In email@example.com,No I didn't, although it might look that way.
> Paul Leyland <paul@...> wrote:
>> I think you're using different vocabularies.
>> In the Cunningham project, and 16^137-1 == 2,548-
>> in their nomenclature, algebraic factors are
>> distinguished from Aurifeuillian factors.
> The problem was that Jane saw Aurifeullian and algebraic
> as being mutually exclusive adjectives. They are not.
What I was doing was objecting to the following, posted earlier in the
> Exercise: Find the algebraic factors of 16^137-1.</quote>
> Comment: The maximum number of digits in any such
> factor is 42, so if your routine produces a larger
> factor it does not know enough algebra.
because, as I showed, there is an alegbraic (composite) factor of
16^137-1 that has 83 (i.e. more than 42) digits.
>Let me make it quite clear: I agree with you, but the statement that the
> The Cunningham project uses this vocabulary:
>> L,M Aurifeuillian algebraic factorization (see III C)
> precisely as I did here:
>> There are two types of algebraic factorization for x^n-1:
>> 1) cyclotomic algebraic factorization
>> 2) Aurifeuillian algebraic factorization
> So at least John Brillhart, D. H. Lehmer, J. L. Selfridge,
> Bryant Tuckerman, and S. S. Wagstaff, Jr agree with me,
> even if Jane or Paul may not :-)
maximum number of digits in any algebraic factor of 16^137-1 is 42 is
wrong, because it's 83.
- --- In firstname.lastname@example.org,
"mikeoakes2" <mikeoakes2@...> wrote:
> > ABC2 if($a>$b,($a^137+$b^137)/($a+$b),0) & ($a^137-$b^137)/($a-$b)It seems that no-one spotted the Aurifeuillian method for
> > a: from 216 to 262
> > b: from 216 to 262
> > does not give a double hit, as Mike has probably checked.
> Your intuition is spot on, as per usual, good Sir:
> they were pfgw jobs of precisely this form that I ran.
> Exercise 4: Find integers (a,b) with min(a,b) > 46225 andAs previously remarked, the trivial Ansatz
> N(a,b) = (a^137 - b^137)/(a - b) equal to the product of
> two primes, each with less than 330 decimal digits.
> Comment: This may solved, systematically, in less than
> 30 seconds, including the time for primality proofs.
(a,b) = (x^2,y^2) cannot solve Exercise 4.
Solution: Aurifeuillian algebraic factorization results from
(a,b) = (x^2,137*y^2), with min(x,sqrt(137)*y) > 215. Since
max(x,sqrt(137)*y) < 10^(329/136) < 263, we have x in the
range [216,262] and y in the range [19,22], yielding a pair
of PRPs in 0.27 seconds and a proof in 26 seconds:
print1([[x,y],[#Str(F),#Str(F)]]" in "gettime" ms. ")))));
if(isprime(F)==[1,1],print("Proof in "round(gettime/10^3)" sec."));}
[[220, 21], [323, 329]] in 270 ms. Proof in 26 sec.
Hence we have the solution N(a,b) = P323*P329,
with (a,b) = (220^2,137*21^2) = (48400,60417).