# Jim White

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## Registered User

 Name Jim White Member for 4 months Seen Feb 20 at 4:46 Website Location Canberra, Australia Age 60
Retired software engineer, now doing computational science research. Special interest in Pell equations, particularly wrt "smooth pairs", ie. the "StÃ¸rmer Problem", and Lehmer's method of solving it.
 Jan22 comment Decision problem wrt Pairs of Polynomials with Integer CoefficientsIf only it were that simple! :) Jan22 asked Decision problem wrt Pairs of Polynomials with Integer Coefficients Jan20 comment Square Roots of Unity modulo N^2BTW, I believe the RHS above should be $-2s\lambda$ Jan18 comment Square Roots of Unity modulo N^2Thank you very much, that helps, and I can see how it extends to $N^2/f^2$. Jan18 comment Square Roots of Unity modulo N^2If that proviso makes this a trivial question, my apologies, and please could I have a reference? Jan18 comment Square Roots of Unity modulo N^2We can assume N is factorable, ie we know all $f|N$. @Felipe, I don't see how Hensel helps with this specific problem. Jan18 revised Square Roots of Unity modulo N^2added 140 characters in body Jan18 asked Square Roots of Unity modulo N^2 Jan16 revised a family of Pellian equationsdeleted 159 characters in body Jan16 revised a family of Pellian equationsdeleted 64 characters in body; deleted 23 characters in body; deleted 86 characters in body Jan16 revised a family of Pellian equationsadded 41 characters in body Jan16 answered a family of Pellian equations Jan15 revised a family of Pellian equationsincorrect formula corrected Jan14 comment a family of Pellian equationsCorrected mistake in the statement "One property shared by all $y_n \in K_1 \ldots$. The divisibility property involves $y_n^2 + 1$, not $y_n$. Jan14 revised a family of Pellian equationscorrected mistake Jan12 revised a family of Pellian equationsadded 3 characters in body Jan12 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. Jan12 revised a family of Pellian equationsadded 154 characters in body Jan12 revised a family of Pellian equationsadded 70 characters in body Jan12 comment a family of Pellian equationsApologies, the bounds for $k$ above should read $2^{32}$ and $2^{64}$ Jan12 comment a family of Pellian equationsMy 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 Jan12 comment a family of Pellian equationsThere 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 $|K_2| = 1,725,632$. Jan12 comment a family of Pellian equationsFor $k < 2^32$ 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 $k$. Jan12 revised a family of Pellian equationsadded 70 characters in body; added 11 characters in body Jan12 revised a family of Pellian equationsadded 962 characters in body Jan12 revised a family of Pellian equationsadded 194 characters in body Jan12 revised a family of Pellian equationsadded 622 characters in body Jan12 revised a family of Pellian equationsadded 191 characters in body; added 8 characters in body Jan12 revised a family of Pellian equationsdeleted 2 characters in body; added 1 characters in body Jan12 revised a family of Pellian equationsadded 1108 characters in body Jan12 revised a family of Pellian equationsadded 657 characters in body; added 2 characters in body; added 3 characters in body Jan12 revised a family of Pellian equationsadded 3 characters in body; added 244 characters in body Jan12 revised a family of Pellian equationsdeleted 14 characters in body; added 138 characters in body Jan12 answered a family of Pellian equations Jan12 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? Jan12 comment Consecutive integers with no large prime factorsYou 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! :-) Jan11 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! Jan11 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 $(pS_1, pS_1 + p). Jan11 comment Greatest Prime Factors: For any prime p, is there an integer C such that for any x >= C, all but 1 integer in the sequence x+1, x+2, …, x+p has a greatest prime factor > p.Correction:$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$you like, so there is no avoiding the complications (wrt the Lehmer method) described above. Jan11 comment Uniqueness of values in recurrence relationsAaron, that sounds like fun, is there anything I can do to contribute? Jan11 comment Uniqueness of values in recurrence relationsThanks Todd! I'm very happy to be here :) Jan11 revised Greatest Prime Factors: For any prime p, is there an integer C such that for any x >= C, all but 1 integer in the sequence x+1, x+2, …, x+p has a greatest prime factor > p.deleted 28 characters in body; added 4 characters in body Jan11 revised Greatest Prime Factors: For any prime p, is there an integer C such that for any x >= C, all but 1 integer in the sequence x+1, x+2, …, x+p has a greatest prime factor > p.added 1 characters in body; added 14 characters in body Jan11 answered Greatest Prime Factors: For any prime p, is there an integer C such that for any x >= C, all but 1 integer in the sequence x+1, x+2, …, x+p has a greatest prime factor > p. Jan10 revised Consecutive integers with no large prime factorsdeleted 1 characters in body Jan10 revised Consecutive integers with no large prime factorsadded 158 characters in body Jan10 answered Consecutive integers with no large prime factors Jan10 comment Uniqueness of values in recurrence relationsAnd the second equation should of course read$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\$. Jan10 awarded ● Commentator Jan10 comment Uniqueness of values in recurrence relationsAnother problem is that you don't seem to be able to edit comments