Jim White
|
Registered User
|
Retired software engineer, now doing computational science research. Special interest in Pell equations, particularly wrt "smooth pairs", ie. the "Størmer Problem", and Lehmer's method of solving it.
|
|
Jan 22 |
comment |
Decision problem wrt Pairs of Polynomials with Integer Coefficients If only it were that simple! :) |
|
Jan 22 |
asked | Decision problem wrt Pairs of Polynomials with Integer Coefficients |
|
Jan 20 |
comment |
Square Roots of Unity modulo N^2 BTW, I believe the RHS above should be $-2s\lambda$ |
|
Jan 18 |
comment |
Square Roots of Unity modulo N^2 Thank you very much, that helps, and I can see how it extends to $N^2/f^2$. |
|
Jan 18 |
comment |
Square Roots of Unity modulo N^2 If that proviso makes this a trivial question, my apologies, and please could I have a reference? |
|
Jan 18 |
comment |
Square Roots of Unity modulo N^2 We can assume N is factorable, ie we know all $f|N$. @Felipe, I don't see how Hensel helps with this specific problem. |
|
Jan 18 |
revised |
Square Roots of Unity modulo N^2 added 140 characters in body |
|
Jan 18 |
asked | Square Roots of Unity modulo N^2 |
|
Jan 16 |
revised |
a family of Pellian equations deleted 159 characters in body |
|
Jan 16 |
revised |
a family of Pellian equations deleted 64 characters in body; deleted 23 characters in body; deleted 86 characters in body |
|
Jan 16 |
revised |
a family of Pellian equations added 41 characters in body |
|
Jan 16 |
answered | a family of Pellian equations |
|
Jan 15 |
revised |
a family of Pellian equations incorrect formula corrected |
|
Jan 14 |
comment |
a family of Pellian equations Corrected mistake in the statement "One property shared by all $y_n \in K_1 \ldots $. The divisibility property involves $y_n^2 + 1$, not $y_n$. |
|
Jan 14 |
revised |
a family of Pellian equations corrected mistake |
|
Jan 12 |
revised |
a family of Pellian equations added 3 characters in body |
|
Jan 12 |
comment |
Greatest Prime Factors: For any prime p, is there an integer C such that for any x >= C, all but 1 integer in the sequence x+1, x+2, …, x+p has a greatest prime factor > p. I ask because I've always been interested in investigating an extension to Lehmer's method as described above, but have had no particular motivation to do so. |
|
Jan 12 |
revised |
a family of Pellian equations added 154 characters in body |
|
Jan 12 |
revised |
a family of Pellian equations added 70 characters in body |
|
Jan 12 |
comment |
a family of Pellian equations Apologies, the bounds for $k$ above should read $2^{32}$ and $2^{64}$ |
|
Jan 12 |
comment |
a family of Pellian equations My co-author Keith Matthews has made available a set of slides he used when giving a recent presentation based on our submitted paper. These can be found at numbertheory.org/pdfs/dujella_slides.pdf |
|
Jan 12 |
comment |
a family of Pellian equations There are too many $k \in K$ to make verification practical for $k < 2^64$,
but for the record we believe that $|K_1| = 3,040,378,747$ and
$|K_2| = 1,725,632$.
|
|
Jan 12 |
comment |
a family of Pellian equations For $k < 2^32$ we have $|K_1| = 48717, |K_2| = 1657, |K| = 50374$.
These were all verified to be unique, so Andrej's conjecture is
confirmed for $k < 2^32$, given that we accept that the enumeration
process finds all $k$.
|
|
Jan 12 |
revised |
a family of Pellian equations added 70 characters in body; added 11 characters in body |
|
Jan 12 |
revised |
a family of Pellian equations added 962 characters in body |
|
Jan 12 |
revised |
a family of Pellian equations added 194 characters in body |
|
Jan 12 |
revised |
a family of Pellian equations added 622 characters in body |
|
Jan 12 |
revised |
a family of Pellian equations added 191 characters in body; added 8 characters in body |
|
Jan 12 |
revised |
a family of Pellian equations deleted 2 characters in body; added 1 characters in body |
|
Jan 12 |
revised |
a family of Pellian equations added 1108 characters in body |
|
Jan 12 |
revised |
a family of Pellian equations added 657 characters in body; added 2 characters in body; added 3 characters in body |
|
Jan 12 |
revised |
a family of Pellian equations added 3 characters in body; added 244 characters in body |
|
Jan 12 |
revised |
a family of Pellian equations deleted 14 characters in body; added 138 characters in body |
|
Jan 12 |
answered | a family of Pellian equations |
|
Jan 12 |
comment |
Greatest Prime Factors: For any prime p, is there an integer C such that for any x >= C, all but 1 integer in the sequence x+1, x+2, …, x+p has a greatest prime factor > p. I'm kind of preoccupied with a couple of other questions, so I don't quite know what you mean by "patterns like A213523". This could just be attention-deficit on my part, but can you explain in more detail what you are looking for? |
|
Jan 12 |
comment |
Consecutive integers with no large prime factors You are absolutely right. Indeed, within Lehmer's paper, it's only in the title that an umlaut is used, within the text proper it's always "Størmer". Apologies to Wiki and anybody else I might have inadvertently offended! :-) |
|
Jan 11 |
comment |
Greatest Prime Factors: For any prime p, is there an integer C such that for any x >= C, all but 1 integer in the sequence x+1, x+2, …, x+p has a greatest prime factor > p. I meant $(pS_1, pS_1 + p)$. You can't edit comments! |
|
Jan 11 |
comment |
Greatest Prime Factors: For any prime p, is there an integer C such that for any x >= C, all but 1 integer in the sequence x+1, x+2, …, x+p has a greatest prime factor > p. Ah, but then again, we might still get lucky, for perhaps our $S_m$ is in fact always $(pS_1, pS_1 + p). |
|
Jan 11 |
comment |
Greatest Prime Factors: For any prime p, is there an integer C such that for any x >= C, all but 1 integer in the sequence x+1, x+2, …, x+p has a greatest prime factor > p. Correction: $S_2 < S_1$ was only true if $gcd(S, S+2)=1$. Clearly any $S, S+1$ will correspond to smooth pairs $(kS, kS + k)$ for any $k$ you like, so there is no avoiding the complications (wrt the Lehmer method) described above. |
|
Jan 11 |
comment |
Uniqueness of values in recurrence relations Aaron, that sounds like fun, is there anything I can do to contribute? |
|
Jan 11 |
comment |
Uniqueness of values in recurrence relations Thanks Todd! I'm very happy to be here :) |
|
Jan 11 |
revised |
Greatest Prime Factors: For any prime p, is there an integer C such that for any x >= C, all but 1 integer in the sequence x+1, x+2, …, x+p has a greatest prime factor > p. deleted 28 characters in body; added 4 characters in body |
|
Jan 11 |
revised |
Greatest Prime Factors: For any prime p, is there an integer C such that for any x >= C, all but 1 integer in the sequence x+1, x+2, …, x+p has a greatest prime factor > p. added 1 characters in body; added 14 characters in body |
|
Jan 11 |
answered | Greatest Prime Factors: For any prime p, is there an integer C such that for any x >= C, all but 1 integer in the sequence x+1, x+2, …, x+p has a greatest prime factor > p. |
|
Jan 10 |
revised |
Consecutive integers with no large prime factors deleted 1 characters in body |
|
Jan 10 |
revised |
Consecutive integers with no large prime factors added 158 characters in body |
|
Jan 10 |
answered | Consecutive integers with no large prime factors |
|
Jan 10 |
comment |
Uniqueness of values in recurrence relations And the second equation should of course read $mz^2 - (m-4)u^2 = 4$. For example, from $29 = U_2(5) = V_2(6)$ we obtain $7v^2 - 3u^2 = 5z^2 - u^2 = 4$ with $z=13, v=19, u=29$. |
|
Jan 10 |
awarded | ● Commentator |
|
Jan 10 |
comment |
Uniqueness of values in recurrence relations Another problem is that you don't seem to be able to edit comments |
