One of my publications is "Jacobi Symbols, ambiguous ideals, and continued fractions", Acta Arith. LXXXV.4 (1998), 331--349. It reflects my interest in the interrelationships between continued fractions and ideal theory. This particular paper generalizes some independent seminal results by Chowla and Schinzel, later developed by Friesen. In order to state some results, let's develop some notation.

Suppose that a=(P+\sqrt(d))/Q is a quadratic irrational, where $d>0 is not a perfect square. Recursively define:

P_0=P,Q_0=Q, and for j>=0:

q_j=\lfloor(P_j+\sqrt(d))/Q\rfloor,

where \lfloor x\rfloor is the greatest integer less than or equal to x,

P_{j+1}=q_jQ_j-P_j,

and

D=P_{j+1}^2+Q_jQ_{j+1}.

Then a=(P_0+\sqrt(d))/Q_0=< q_0;q_1,...,q_j,...>

is the simple continued fraction expansion of a.

With this in mind, we may now state a sample result.

Theorem

Suppose that D=4d is a discriminant with radicand d=ab congruent to 1 modulo 4 where a is congruent to 3 modulo 4, (a,b\in N). Also, let L denote the period length of the simple continued fraction expansion of \sqrt(d). If L is even and a=Q_{L/2}, then

(a/b)=(-1)^{L/2} and (b/a)=(-1)^{L/2+1},

where (*/*) is the Jacobi symbol.

An immediate consequence of this is a result by Friesen, which in turn had generalized a result by Chowla.

Corollary

If d=pq, where p and q are prime both congruent to 3 modulo 4, and p is smaller than q, then the following Legendre symbol equalities hold:

(p/q)=(-1)^{L/2} and (q/p)=(-1)^{L/2+1}.

There are sveral results of this nature in the paper together with related results on the alternating sums:

sigma=(1)^{L}q_0+(-1)^{L-1}q_1+(-1)^{L-2}q_2+...+(-1)q_{L-1}+q_L.

A paper coauthored with Andrew Granville (Acta Arith. XCVI.2 (2000), 139--153. download via: ) deals with another of my passion prime-producing quadratic polynomials. Some comments on the results of this paper are as follows:

Rabinowitsch, at the 1912 International Congress of Mathematicians, showed that n^2+n+A is prime for n=0,1,2,... A-2 if and only if 4A-1 is squarefree and the ring of integers of the field Q(\sqrt{1-4A}) has just one equivalence class of ideals. One can generalize Rabinowitsch's criterion to other polynomials, and to other fields; for example, Hugh Williams and I proved the following, which Granville has dubbed the "Rabinowitsch-Mollin-Williams" criterion for real quadratic fields:

Theorem

n^2+n-A is prime for all positive n< \sqrt{A}-1 if and only if the field Q(\sqrt{4A+1}) has class number one where either A=4, or A is odd, bigger than 5 is odd and is of the form m^2 or m^2+m\pm 1 for some integer m, see my Theorem 6.5.13 on page 352 of my book Fundamental Number Theory with Applications, cited at the bottom of this page.

One can develop similar criterion for when the class number is 2, or 3, or any fixed number (see my paper "Prime-producing quadratics" Amer. Math. Monthly, 104, (1997), 529--544 (download via ), and also my book Quadratics cited at the bottom of this page).

The idea in all of these proofs is that if a large proportion of the values of a quadratic polynomial of discriminant d are prime then there cannot be many small primes p for which (d/p)=1 (else those small primes would divide the values of the given quadratic polynomial, preventing it from being prime very often). If that is the case then the value of the L-function L(1,(d/.)) will be surprisingly small, which is equivalent to having h(d), the class number, small if d<0, and to having both h(d) and \epsilon_d, the fundamental unit, small if d>0. We remark that \epsilon_d is ``small'' if and only if the continued fraction for (1+\sqrt{d})/2 or \sqrt{d}/2 (as d is congruent to 1 or 0 modulo 4) is short, that is if d is a value of one of several special forms.

Siegel showed that L(1,(d/.))>> 1/d^{o(1)}$ and Tatuzawa made Siegel's argument explicit, excluding at most one d, a presumably hypothetical counterexample to the Generalized Riemann Hypothesis. Using Tatuzawa's result, I was able to give gives many explicit criteria ``with one possible exception'' (see my book Quadratics).

One might ask whether it is possible to find quadratic polynomials with arbitrarily long strings of consecutive prime values; that is whether, for any given N can we find A for which n^2+n+A is prime for n=0,1,2... N? This is an open question, though in the paper we show that such polynomials exist assuming the "prime k-tuplets conjecture".

In this paper we are primarily interested in further developing the theory of quadratic polynomials for which many of the small values are prime (rather than ``all'' as in Rabinowitsch's result). It is well-known that if the class number of some imaginary quadratic field with large discriminant is one then we will have an egregious counterexample to the Generalized Riemann Hypothesis (that is, a zero of the associated Dirichlet L-function which is very close to 1, a weak consequence of the Generalized Riemann Hypothesis). Thus Rabinowitsch's result can be informally stated as ``n^2+n+A is prime for n=0,1,2,...A-2 and A>41 if and only if the Generalized Riemann Hypothesis is very badly false for some quadratic Dirichlet L-function''. One might guess that if n^2+n+A$is prime for very many of the numbers n=0,1,2,... A-2 (though not all) then perhaps still the Generalized Riemann Hypothesis is false, though perhaps not with a zero quite so close to 1. This is indeed the case:

Proposition

There exists a constant k_1>0 such that if there are more than k_1 N \log\log |A|/\log N primes amongst the integers n^2+A or n^2+n+A for n=0,1,2\dots N for some N then the Generalized Riemann Hypothesis is false.

A sample main result from our paper that gives the flavour of the other results is the following:

Theorem

For large R and N in the range R^\epsilon less than N less than R^{1/2} we have:

# { n<= N: n^2+n+A is prime} is asymptotic to N/log N L(1,((1-4A)/.))^{-1}

for at least a positive proportion of the integers A in the range R less than A less than 2R.

Another paper, coauthored with Hugh Williams and Andrew Granville: An upper bound on the last inert prime in a real quadratic field, Canad. Math. J. 52 (2000), 369--380, has as its main result of the paper is the proof of a conjecture of mine cited on page 220 of Quadratics. I had always been interested in this problem, and could prove it under the assumption of the generalized Rieman hypothesis. However, I am happy to say that it is now solved unconditionally as follows.

It is shown by a combination of analytic and computational techniques that for any positive fundamental discriminant D > 3705, there is always at least one prime p < \sqrt{D}/2 such that the Kronecker symbol (D/p) = -1.

The complete explicit list of those D<=3705, for which there are no inert primes less than the Minkowski bound are given on the aforementioned page of Quadratics.

A recent topic of interest is construction of infinite families of Pellian (and related) polynomials from the perspective of their continued fraction expansions. This is a generalization of the notion of RD types that have been so completely studied (see chapter 3 of my book Quadratics cited below for a complete background on RD types). Also, for this new approach that generalizes RD types, see paper: (download via ). Several other papers on the topic are (from my list of publications) # 135, # 141-142, #144-145, #148, #151 and #154.

Another topic of recent interest is to look at the central norm (that value Q_j as defined above) where j is half the period length of the continued fraction when the peiod length is even. I have completely solved the problem as to when that value is a power of 2, for instance, and looked at other questions surrounding this issue (see publications # 152 and #153, for instance).

Another topic related to Diophantine analysis is my work on the equation ax^2-by^2=+-1. I approach this from a continued fraction perspective and solve problems as well as re-cast classical problems in this light (see publications #139, # 146, and # 153 for recent results). However, the most comprehensive paper in this direction is given by #155, which may be downloaded via , which appeared in 2004 (see)

A paper on the Tate-Shafarevich groups is # 150, for those interested in that topic (which may be downloaded via ). There is also recent work on developing infinitely many Diophantine equations 2x^2-cy^2=-1 which are NOT solvable for any integers x, y; yet 2x^2-cy^2 =-1 modulo n is solvable for every natural number n>1, see paper: (download via ), which appeared in JP Journal of Algebra, Number Theory, and Applications, Vol. 4 (2004), 353--362.

Another paper deals with the number of solutions of the Diophantine equation Dx^2+E=k^n, which contains counterexamples to some of the results in the recent paper in Crelle's journal (volume 539 (2001), 55--74) by Bugeaud and Shorey: On the number of solutions of the generalized Ramanujan-Nagell equation. This is different from the counterexample found by Leu and Li in their Proceedings of the AMS paper (Volume 131, pp. 3643--3645): The Diophantine equation 2x^2+1=3^n. I am looking at this from a continued fraction perspective and show that there are easy ways to get the solutions that have not been considered in the literature. Moreove, papers such as the Leu-Li paper mentioned above, are really footnotes of work by Ljunggren in the early 1940s. To get a copy of my paper, download via ).

More recently I have been looking at generalizing criteria given by Lagrange involving central norms. See: Generalized Lagrange Criteria for Certain Quadratic Diophantine Equations (2005)

My latest book is: CODES --- The Guide to Secrecy from Ancient to Modern Times (Chapman and Hall/CRC - Taylor and Francis Group (ISBN #1-58488-470-3)

There are numerous other new research interests that I will add here at a later date. There is much yet to be done.

Also, see my undergraduate text:

Fundamental Number Theory with Applications (CRC PRESS 1998:ISBN #0-8493-3987-1),

as well as my senior undergrad./beginning grad. text:

Algebraic Number Theory (CHAPMAN and HALL/CRC PRESS:ISBN #0-8493-3989-8).

For info on my past research interests see my upper graduate text:

QUADRATICS (CRC PRESS 1995:ISBN #0-8493-3983-9).

For info on my cryptographic interests, see my texts:

AN INTRODUCTION TO CRYPTOGRAPHY (CHAPMAN and HALL/CRC PRESS 2000:ISBN# 1-58488-127-5)

which is going to its second edition in the Fall of 2006, and

RSA and Public-Key Cryptography (CHAPMAN and HALL/CRC PRESS 2003:ISBN# 1-58488-338-3).

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Last updated: August 24, 2006

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