I've finished a book on SAGE and differential calculus. The idea was to simply take the old book Granville's Elements of Calculus, which is public domain now, and slightly modernize and "SAGE-ify" it. The source code, with the pdf form (and the pdf scan of the original book of Granville) can be found at
http://www.opensourcemath.org/books/granville-calculus/ or
http://sage.math.washington.edu/home/wdj/teaching/.
It is open-source and about 230+ pages printed. Hope it is useful for some people.
It is not quite in it's final form, so if you find typos, of have comments, please feel free to email me.
Wednesday, January 30, 2008
Wednesday, January 2, 2008
first aliquot factor found by SAGE ECM interface
Paul Zimmermann (http://www.loria.fr/~zimmerma/records/aliquot.html) reports:
The SAGE ECM interface found a first factor of the aliquot sequence starting
from 552:
remains 23648161798622140141259448258749760352819524456141488104537419990481892694930432002158957619604181055633215274583954462907657503167424176909 (140 digits)
found factor by ecm: 58417195751812372006463994075468288063413 with parameters {'poly': 'Dickson(6)', 'sigma': '300411371', 'B1': '3990569', 'B2': '8561602150'}
Other nice factors will surely follow.
An aliquot sequence is simply the iteration of the function n -> sigma(n)-n,
where sigma(n) is the "sum of divisors" function. One open question from
Catalan is whether this sequence always converges to 1 (or to a cycle). The
first to perform extensive computations on aliquot sequences was Lehmer, who
found that all sequences starting from n <= 1000 converge, except perhaps
n=276, 552, 564, 660 and 966. These are the "Lehmer five" sequences. Since
several years, together with other people, I try to extend these Lehmer five
sequences. The main difficulty is that to compute sigma(n), you have to
factor n. For the current large numbers we encounter (150-160 digits) we use
a combination of different algorithms (ECM, QS, NFS). I have now converted
to SAGE the script that (tries to) extend aliquot sequences. The above
factorization is a first success of the new script.
The SAGE ECM interface found a first factor of the aliquot sequence starting
from 552:
remains 236481617986221401412594482587
found factor by ecm: 584171957518123720064639940754
Other nice factors will surely follow.
An aliquot sequence is simply the iteration of the function n -> sigma(n)-n,
where sigma(n) is the "sum of divisors" function. One open question from
Catalan is whether this sequence always converges to 1 (or to a cycle). The
first to perform extensive computations on aliquot sequences was Lehmer, who
found that all sequences starting from n <= 1000 converge, except perhaps
n=276, 552, 564, 660 and 966. These are the "Lehmer five" sequences. Since
several years, together with other people, I try to extend these Lehmer five
sequences. The main difficulty is that to compute sigma(n), you have to
factor n. For the current large numbers we encounter (150-160 digits) we use
a combination of different algorithms (ECM, QS, NFS). I have now converted
to SAGE the script that (tries to) extend aliquot sequences. The above
factorization is a first success of the new script.
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