Mike Knapp - Papers and Preprints

Note: The versions of these papers that I have put here are preprints. They may differ from the final published version.

In Preparation

[1] (with C. Broll, J. Jennings, P.H.A. Rodrigues and D. Veras) More exact values of the function Gamma*(k).

[2] (with H. Godinho and B. Miranda) Sextic forms over Q_2(\sqrt{5}).

Submitted

[3] Inhomogeneous additive equations 24pp. (submitted.) .

a1x1k + ... + asxsk + b1y1n + ... + bsysn = 0,
where all the coefficients are integers. The main result of the article is to give a sharp upper bound on the smallest value of s (as a function of k and n) that guarantees that this equation has nontrivial solutions in the p-adic field Qp for all primes p. We also give the exact smallest value of s for various choices of k and n.

[4] Integral values of generating functions related to the Fibonacci sequence 19pp. (submitted.)

Abstract: Let a be an integer greater than 2, and define a sequence recursively by G0=a, G1=1, and Gn = Gn-1 + Gn-2 for all integers n. Let G(x) be the generating function for this sequence. In this article, we give a method to find all rational values of x such that G(x) is an integer, and give several families of explicit solutions. For many values of the parameter a, these solutions are the only ones.

Accepted for Publication / In Print

[5] (with A. Cambraia Jr., A. Lemos, B. K. Moriya, and P. H. A. Rodrigues) On prime factors of Mersenne numbers, Palestine Journal of Mathematics (to appear).

Abstract: The Mersenne numbers are numbers of the form Mn = 2n - 1, where n is a positive integer. The main theorem in this article is to give a description of all the Mersenne numbers which have at most 3 distinct prime factors. We also show that it is rare for a Mersenne number to have few prime factors, and additionally study the equation Mm + Mn = 2pa, where p is a prime number.

[6] (with H. Godinho) Infinitely many counterexamples to a conjecture of Norton. Michigan Math. J. 69 (2020), 533-543.

Abstract: The number Γ(k) is defined as the least integer s such that the diagonal form a1x1k + ... + asxsk is guaranteed to have a nontrivial zero in every p-adic field Qp, regardless of the coefficients. An old conjecture of Norton posited that Γ(k) ≡ 1 (mod k) for all k. Although this conjecture has long been known to be false, only 3 counterexamples have been produced, with two of them only coming in recent years. In this article, we produce infinitely many values of k for which Norton's conjecture is false.

[7] Distance pebbling on directed cycle graphs. J. Combin. Math. Combin. Comput. 111 (2019), 283-286.

Abstract: In this article, we answer the following question about graph pebbling. Consider a cycle graph with n vertices and all edges directed counterclockwise, and choose a positive integer d with d < n. We determine the smallest number s such that if s pebbles are placed on the graph, then there is at least one pebble which can be moved a distance of at least d vertices away from its original position through a sequence of pebbling moves.

[8] (with H. Godinho, P. H. A. Rodrigues, and D. Veras) On the values of Gamma^*(k,p) and Gamma^*(k). ACTA Arith. 191.1 (2019), 67-80.

Abstract: In this article, we calculate an upper bound for the value of Gamma^*(k,p), which gives the smallest number of variables necessary to guarantee that any additive form of degree k with integer coefficients has a p-adic solution, and also give a condition under which this upper bound is actually an equality. Additionally, we calculate the value of Gamma^*(54), which is the smallest number of variables needed to guarantee that any additive form of degree 54 with integer coefficients has a zero in every p-adic field Qp.

[9] Solubility of additive sextic forms over unramified quadratic extensions of Q_2. Publ. Math. Debrecen 95/1-2 (2019), 67-91.

Abstract: Consider an additive form a1x16 + ... + asxs6 whose coefficients are integers of one of the six unramified quadratic extensions of the p-adic field Q2. In this article, we show that if the form has at least 9 variables, then it must have a nontrivial zero. For four of these extensions, we show that no smaller number of variables has this property.

[10] 2-Adic zeros of diagonal forms. Journal of Number Theory 193C (2018), 37-47.

Abstract: Consider an additive form a1x1d + ... + asxsd whose coefficients are all 2-adic integers. In this article we give an exact formula, in terms of d, for the smallest number of variables which guarantees that this form has a nontrivial zero in the 2-adic integers regardless of the values of the coefficients.

[11] (With R. Auer) Studying baseball's wild-card team using probability Mathematics Teacher 107 (2013), No. 1, August 2013, 74-77. (This is a Microsoft Word document)

Abstract: Major League Baseball's decision more than a decade ago to add wild card teams to its playoff structure remains controversial even now. In this article, we attempt to support the notion that there should be wild card teams. To do this, we make a very simple probabalistic model of Major League Baseball's final regular-season standings. Then we use this model to predict how often the team that becomes the wild card will have a better record than the worst division-winning team, and calculate the expected rank in the league of the worst division winner. Perhaps surprisingly, our model predicts that the worst division winner should finish 7th or worse in the league more than 12% of the time.

[12] (With A. Fox) A note on weakly complete sequences. Journal of Integer Sequences 16 (2013), no. 6, Article 13.6.2, 14pp.

Abstract: In this article, we treat sequences formed by the same procedure as the ones in the article "Weakly complete sequences formed by the greedy algorithm" below. We show that the patterns described in that abstract will always exist whenever 1 < d < n-7. This majority of this work was done by Alyson Fox while she was my student in the Hauber summer research program at Loyola.

[13] (With H. Godinho and P. H. A. Rodrigues) Pairs of additive sextic forms. J. Number Theory 133 (2013), 176-194.

Abstract: This article shows that a system of two homogeneous additive polynomials of degree 6 with integer coefficients will have a nontrivial p-adic solution for any prime p provided only that the system has a total of 73 variables.

[14] Pairs of additive forms of odd degrees. Michigan Math J. 61 (2012), 493-505.

Abstract: Suppose that k and n are odd integers. We show that any system of diagonal forms of degrees k and n, with integer coefficients, in at least k2+n2+1 variables has a nontrivial zero in the field Qp for each prime p.

[15] (With M. Paul) On weakly complete sequences formed by the greedy algorithm. Integers 12 (2012) A4, 17pp.

Abstract: Consider a sequence defined recursively by the following procedure. Pick numbers n and d, and set a1=n and a2=n+d. Then for each k>2, set ak to be the smallest number greater than ak-1 which cannot be written as a sum of previous terms of the sequence. Experimentation shows that the terms of the sequence resulting from this procedure appear to have several very interesting patterns. We show that these patterns really exist in the cases where n=1, or where d=1, or where n=d. The majority of this work was done by Michael Paul while he was my student in the Hauber summer research program at Loyola.

[16] Exact values of the function Gamma*(k). J. Number Theory 131 (2011), 1901-1911.

Abstract: The function Gamma*(k) gives the smallest number of variables required to ensure that an additive form of degree k has a nontrivial solution in p-adic integers for every prime p. In this article, we evaluate Gamma*(k) exactly when k=14, 20, 24, 26, 27, 29 and 31. With these results, the exact value of Gamma*(k) is now known whenever k < 32.

[17] (With K. Mohamed and Z. Tahar) Hensel codes of square roots of p-adic numbers. Appl. Anal. Discrete Math. 4 (2010), 32-44. [PDF]

Abstract: Suppose that a is a p-adic number. In this paper, we consider the problem of calculating the p-adic digits of the square root of a. We do this using two fixed-point methods with quadratic and cubic rates of convergence. For each method, we calculate the minimum number of correct digits obtained after n iterations, and the number of iterations required to obtain an approximation of any given accuracy.

[18] (With C. Xenophontos) Numerical analysis meets number theory: using rootfinding methods to calculate inverses mod pn. Appl. Anal. Discrete Math. 4 (2010), 23-31. [PDF]

Abstract: In this expository paper, we show how rootfinding methods from numerical analysis (Newton's method and the secant method) can be used to calculate inverses of numbers modulo powers of primes. In undergraduate courses, the fields of numerical analysis and number theory can appear to be unrelated, and we believe that students may enjoy seeing a connection between them.

[19]Sines and cosines of angles in arithmetic progression Math. Mag. 82 (2009), 371-372. [PDF] [PS] [DVI]

Abstract: In this expository paper, we derive a formula for the sum of a series of sines or cosines, where the angles form an arithmetic progression. This derivation of the formula was originally given in the journal Arbelos, which is now sadly out of print. The derivation is interesting because it does not make any use of complex numbers.

[20]On systems of diagonal forms, II. Math. Proc. Cambridge Philos. Soc. 147 (2009), 31-45. [PDF] [PS] [DVI]

Abstract: This paper is similar to [3] below, except that we are able to remove the restriction that the degrees of the forms must be all different. Given a system of diagonal forms over Qp, we ask how many variables are required to guarantee that the system has a nontrivial zero. We show that if the prime p satisfies p > (largest degree) - (smallest degree) + 1, then there is a bound on the sufficient number of variables which is a polynomial in the degrees of the forms. A result of Lewis & Montgomery implies that any bound which works for all primes must exhibit exponential growth. So this result can be thought of as bounding how small p can be before exponential growth is required.

[21]Pairs of homogeneous additive equations. Bull. London Math. Soc. 40 (2008), 447-456 [PDF] [PS] [DVI]
(Note: The PDF files are the final version of this paper from the LMS website. The PS and DVI versions are preprints, and may differ somewhat from the final version.)

Abstract: In 1966, Davenport & Lewis published their paper Notes on congruences III, in which they proved that under some mild conditions, a system of two additive forms must have a nonsingular simultaneous zero modulo any prime number. In their paper, they ask whether the theorem is true in general finite fields, and point out that one of their key lemmas is no longer true in this situation. In this paper we answer their question in the affirmative, proving that under the same conditions, a system of two additive forms over a finite field must have a nonsingular simultaneous zero. We then apply this result to obtain an upper bound on the number of variables required to ensure that a system of two additive forms of equal degrees has a nontrivial zero in a p-adic field.

[22]On systems of diagonal forms. J. Aust. Math. Soc. 82 (2007), 221-236 [PDF] [PS] [DVI]

Abstract: It is known that any system of diagonal forms has a nontrivial p-adic zero for all primes p provided that the number of variables is large enough in terms of the degrees. A result of Lewis & Montgomery shows that the required number of variables must exhibit exponential growth. However, a theorem of Ax & Kochen states that if p is large enough, then a bound which exhibits only polynomial growth suffices. The purpose of this paper is to bound the primes for which exponential growth is required in the situation where the degrees are all different. In particular, if the degrees of the forms are k1 > ... >kR and if p > k1-kR+1, then we give a bound for the required number of variables which has only polynomial growth. This bound is larger than the Ax-Kochen bound, but we note that it applies for primes which are smaller than the largest of the degrees.

[23] Diagonal equations of different degrees over p-adic fields. Acta Arith. 126 (2007), 139-154. [PDF] [PS] [DVI]

Abstract: In this paper we develop a bound on the number of variables required to guarantee that two diagonal homogeneous polynomials of different degrees k and n with coefficients in Qp have a nontrivial simultaneous zero in p-adic integers. If k and n are large and close together, then this is the best known bound. We also prove a much stronger bound in the case when at least one of k and n is odd.

[24]Two by two matrices with both eigenvalues in Z/pZ. Math. Mag. 79 (2006), 145-157.[PDF] [PS] [DVI]

Abstract: A recent paper in Math Magazine gave a proof that there are exactly p2(p2+2p-1)/2 two-by-two matrices with entries in Z/pZ and both eigenvalues also in this field. The proof appeals to a theorem from abstract algebra in the key step. In this note, we give a simpler proof that does not require abstract algebra, using only concepts which could be taught in an undergraduate number theory course.

[25] Artin's conjecture for forms of degree 7 and 11. J. London Math. Soc. (2) 63 (2001), 268-274. [PDF] [PS] [DVI]

Abstract: It is known by work of Ax & Kochen that given any natural number d, any homogeneous polynomial of degree d in d2+1 variables with coefficients in Qp has a nontrivial zero provided that p is sufficiently large. In this paper, we give upper bounds on how large p needs to be when we have either d=7 or d=11.

[26] Systems of diagonal equations over p-adic fields. J. London Math. Soc. (2) 63 (2001), 257-267. [PDF] [PS] [DVI]

Abstract: The main result of this paper is that any system of R diagonal (additive) homogeneous polynomials of degree k in at least 4R2k2 variables with coefficients in Qp has a nontrivial simultaneous zero in p-adic integers. This improves on work of Brudern & Godinho, and is the best known bound when k is even and suitably large in comparison to R. A version of this theorem is also developed for the situation in which Qp is replaced by a finite extension of Qp.

[27] (With P. Isihara) Basic Z12 analysis of musical chords. UMAP J. 14 (1993), 319-348.

Abstract: In this expository paper, we introduce the reader to the analysis of musical chords using the additive group Z/12Z and the twelve-tone operators, which can be thought of as functions on Z/12Z.

Ph.D. Thesis

[28] Forms in many variables over p-adic fields. Ph.D. thesis, University of Michigan, 2000. [PDF] [PS] [DVI]

Abstract: This is my Ph.D. thesis. The content includes that of papers [4] and [5] above, and also some of [1]. Additionally, we elaborate on work of Skinner to give a bound on the number of variables necessary to guarantee that there exists a nontrivial simultaneous zero of a system of R homogeneous additive polynomials of degree pt with coefficients in a finite extension of Qp. This bound does not depend on the degree of the field extension. There is also an introduction to the field of study of zeros of homogeneous equations over p-adic fields.