Journal articles: 'Arithmetic Progression'

1

Ginat, David. "No arithmetic progression." ACM Inroads 5, no. 3 (September 5, 2014): 42–43. http://dx.doi.org/10.1145/2655759.2655772.

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Dinneen, Michael J., Nan Rosemary Ke, and Masoud Khosravani. "Arithmetic Progression Graphs." Universal Journal of Applied Mathematics 2, no. 8 (October 2014): 290–97. http://dx.doi.org/10.13189/ujam.2014.020803.

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3

Sim, Kai An, and Kok Bin Wong. "Magic Square and Arrangement of Consecutive Integers That Avoids k-Term Arithmetic Progressions." Mathematics 9, no. 18 (September 14, 2021): 2259. http://dx.doi.org/10.3390/math9182259.

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In 1977, Davis et al., proposed a method to generate an arrangement of [n]={1,2,…,n} that avoids three-term monotone arithmetic progressions. Consequently, this arrangement avoids k-term monotone arithmetic progressions in [n] for k≥3. Hence, we are interested in finding an arrangement of [n] that avoids k-term monotone arithmetic progression, but allows k−1-term monotone arithmetic progression. In this paper, we propose a method to rearrange the rows of a magic square of order 2k−3 and show that this arrangement does not contain a k-term monotone arithmetic progression. Consequently, we show that there exists an arrangement of n consecutive integers such that it does not contain a k-term monotone arithmetic progression, but it contains a k−1-term monotone arithmetic progression.

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4

Bremner, Andrew, and Samir Siksek. "Squares in arithmetic progression over cubic fields." International Journal of Number Theory 12, no. 05 (May 10, 2016): 1409–14. http://dx.doi.org/10.1142/s179304211650086x.

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Euler showed that there can be no more than three integer squares in arithmetic progression. In quadratic number fields, Xarles has shown that there can be arithmetic progressions of five squares, but not of six. Here, we prove that there are no cubic number fields which contain five squares in arithmetic progression.

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Sanna, Carlo. "Covering an arithmetic progression with geometric progressions and vice versa." International Journal of Number Theory 10, no. 06 (August 14, 2014): 1577–82. http://dx.doi.org/10.1142/s1793042114500456.

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We show that there exists a positive constant C such that the following holds: Given an infinite arithmetic progression [Formula: see text] of real numbers and a sufficiently large integer n (depending on [Formula: see text]), there is a need of at least Cn geometric progressions to cover the first n terms of [Formula: see text]. A similar result is presented, with the role of arithmetic and geometric progressions reversed.

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Buchholz, R. H., and J. A. MacDougall. "Heron quadrilaterals with sides in arithmetic or geometric progression." Bulletin of the Australian Mathematical Society 59, no. 2 (April 1999): 263–69. http://dx.doi.org/10.1017/s0004972700032883.

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We study triangles and cyclic quadrilaterals which have rational area and whose sides form geometric or arithmetic progressions. A complete characterisation is given for the infinite family of triangles with sides in arithmetic progression. We show that there are no triangles with sides in geometric progression. We also show that apart from the square there are no cyclic quadrilaterals whose sides form either a geometric or an arithmetic progression. The solution of both quadrilateral cases involves searching for rational points on certain elliptic curves.

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7

MacDougall, Jim. "79.45 Some Arithmetic Progression Identities." Mathematical Gazette 79, no. 485 (July 1995): 390. http://dx.doi.org/10.2307/3618327.

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8

Heule, Marijn J. H. "Avoiding triples in arithmetic progression." Journal of Combinatorics 8, no. 3 (2017): 391–422. http://dx.doi.org/10.4310/joc.2017.v8.n3.a1.

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9

Bagemihl, Frederick, and F. Bagemihl. "ORDINAL NUMBERS IN ARITHMETIC PROGRESSION." Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 38, no. 1 (1992): 525–28. http://dx.doi.org/10.1002/malq.19920380148.

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10

Bourgain, J. "On Triples in Arithmetic Progression." Geometric And Functional Analysis 9, no. 5 (December 1, 1999): 968–84. http://dx.doi.org/10.1007/s000390050105.

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11

Grynkiewicz, David J., Andreas Philipp, and Vadim Ponomarenko. "Arithmetic-progression-weighted subsequence sums." Israel Journal of Mathematics 193, no. 1 (September 20, 2012): 359–98. http://dx.doi.org/10.1007/s11856-012-0119-8.

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12

Croot, Ernie. "The Minimal Number of Three-Term Arithmetic Progressions Modulo a Prime Converges to a Limit." Canadian Mathematical Bulletin 51, no. 1 (March 1, 2008): 47–56. http://dx.doi.org/10.4153/cmb-2008-006-9.

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AbstractHow few three-term arithmetic progressions can a subset S ⊆ ℤN := ℤ/Nℤ have if |S| ≥ υN (that is, S has density at least υ)? Varnavides showed that this number of arithmetic progressions is at least c(υ)N2 for sufficiently large integers N. It is well known that determining good lower bounds for c(υ) > 0 is at the same level of depth as Erdös's famous conjecture about whether a subset T of the naturals where Σn∈T 1/n diverges, has a k-term arithmetic progression for k = 3 (that is, a three-term arithmetic progression).We answer a question posed by B. Green about how this minimial number of progressions oscillates for a fixed density υ as N runs through the primes, and as N runs through the odd positive integers.

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13

ALVARADO, ALEJANDRA, and EDRAY HERBER GOINS. "ARITHMETIC PROGRESSIONS ON CONIC SECTIONS." International Journal of Number Theory 09, no. 06 (September 2013): 1379–93. http://dx.doi.org/10.1142/s1793042113500322.

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The set {1, 25, 49} is a 3-term collection of integers which forms an arithmetic progression of perfect squares. We view the set {(1, 1), (5, 25), (7, 49)} as a 3-term collection of rational points on the parabola y = x2 whose y-coordinates form an arithmetic progression. In this exposition, we provide a generalization to 3-term arithmetic progressions on arbitrary conic sections [Formula: see text] with respect to a linear rational map [Formula: see text]. We explain how this construction is related to rational points on the universal elliptic curve Y2 + 4XY + 4kY = X3 + kX2 classifying those curves possessing a rational 4-torsion point.

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14

Ferreira, Luis Dias. "Arithmetic Triangle." Journal of Mathematics Research 9, no. 2 (March 21, 2017): 100. http://dx.doi.org/10.5539/jmr.v9n2p100.

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The product of the first $n$ terms of an arithmetic progression may be developed in a polynomial of $n$ terms. Each one of them presents a coefficient $C_{nk}$ that is independent from the initial term and the common difference of the progression. The most interesting point is that one may construct an "Arithmetic Triangle'', displaying these coefficients, in a similar way one does with Pascal's Triangle. Moreover, some remarkable properties, mainly concerning factorials, characterize the Triangle. Other related `triangles' -- eventually treated as matrices -- also display curious facts, in their linear \emph{modus operandi}, such as successive "descendances''.

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15

Mukhopadhyay, Anirban, and T. N. Shorey. "Almost squares in arithmetic progression (II)." Acta Arithmetica 110, no. 1 (2003): 1–14. http://dx.doi.org/10.4064/aa110-1-1.

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16

Mikami, Kentaro, Jun O'Hara, and Kunio Sugawara. "Triangles with sides in arithmetic progression." Elemente der Mathematik 72, no. 2 (2017): 75–79. http://dx.doi.org/10.4171/em/327.

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17

Varbanec, P. D., and P. Zarzycki. "Divisors of Integers in Arithmetic Progression." Canadian Mathematical Bulletin 33, no. 2 (June 1, 1990): 129–34. http://dx.doi.org/10.4153/cmb-1990-022-8.

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AbstractLet d(n; l, k) be the number of positive divisors of n which lie in the arithmetic progression l mod k. Using the complex integration technique the formula is proved. This formula holds uniformly in l, k and x satisfying 1 ≦ l ≦ k, (lx)1/2 ≦ k ≦ x1-∊; the exponent α ≦ 1/3.

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18

Hajdu, L., and T. Kovács. "Almost fifth powers in arithmetic progression." Journal of Number Theory 131, no. 10 (October 2011): 1912–23. http://dx.doi.org/10.1016/j.jnt.2011.04.009.

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19

González-Jiménez, Enrique, and José M. Tornero. "Markoff–Rosenberger triples in arithmetic progression." Journal of Symbolic Computation 53 (June 2013): 53–63. http://dx.doi.org/10.1016/j.jsc.2012.11.003.

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20

Konvalina, J., and Y. H. Liu. "Arithmetic progression sums of binomial coefficients." Applied Mathematics Letters 10, no. 4 (July 1997): 11–13. http://dx.doi.org/10.1016/s0893-9659(97)00051-7.

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21

Chan, Tsz Ho. "Squarefull numbers in arithmetic progression II." Journal of Number Theory 152 (July 2015): 90–104. http://dx.doi.org/10.1016/j.jnt.2014.12.019.

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Mukhopadhyay, Anirban, and T. N. Shorey. "Almost squares in arithmetic progression (III)." Indagationes Mathematicae 15, no. 4 (2004): 523–33. http://dx.doi.org/10.1016/s0019-3577(04)80016-8.

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23

Pritchard, Paul A., Andrew Moran, and Anthony Thyssen. "Twenty-two primes in arithmetic progression." Mathematics of Computation 64, no. 211 (September 1, 1995): 1337. http://dx.doi.org/10.1090/s0025-5718-1995-1297475-1.

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24

Dubner, Harvey, and Harry Nelson. "Seven consecutive primes in arithmetic progression." Mathematics of Computation 66, no. 220 (October 1, 1997): 1743–50. http://dx.doi.org/10.1090/s0025-5718-97-00875-2.

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25

Saradha, N., and T. N. Shorey. "Almost perfect powers in arithmetic progression." Acta Arithmetica 99, no. 4 (2001): 363–88. http://dx.doi.org/10.4064/aa99-4-5.

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26

Dubner, H., T. Forbes, N. Lygeros, M. Mizony, H. Nelson, and P. Zimmermann. "Ten consecutive primes in arithmetic progression." Mathematics of Computation 71, no. 239 (November 28, 2001): 1323–28. http://dx.doi.org/10.1090/s0025-5718-01-01374-6.

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27

Bianchi, Mariagrazia, Stephen P. Glasby, and Cheryl E. Praeger. "Conjugacy class sizes in arithmetic progression." Journal of Group Theory 23, no. 6 (November 1, 2020): 1039–56. http://dx.doi.org/10.1515/jgth-2020-0046.

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AbstractLet {\mathrm{cs}(G)} denote the set of conjugacy class sizes of a group G, and let \mathrm{cs}^{*}(G)=\mathrm{cs}(G)\setminus\{1\} be the sizes of non-central classes. We prove three results. We classify all finite groups for which (1) {\mathrm{cs}(G)=\{a,a+d,\dots,a+rd\}} is an arithmetic progression with {r\geqslant 2}; (2) {\mathrm{cs}^{*}(G)=\{2,4,6\}} is the smallest case where {\mathrm{cs}^{*}(G)} is an arithmetic progression of length more than 2 (our most substantial result); (3) the largest two members of {\mathrm{cs}^{*}(G)} are coprime. For (3), it is not obvious, but it is true that {\mathrm{cs}^{*}(G)} has two elements, and so is an arithmetic progression.

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28

Hajdu, Lajos, and Szabolcs Tengely. "Powers in arithmetic progressions." Ramanujan Journal 55, no. 3 (January 30, 2021): 965–86. http://dx.doi.org/10.1007/s11139-020-00331-5.

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AbstractWe investigate the function $$P_{a,b;N}(\ell )$$ P a , b ; N ( ℓ ) describing the number of $$\ell $$ ℓ -th powers among the first N terms of an arithmetic progression $$ax+b$$ a x + b . We completely describe the arithmetic progressions containing the most $$\ell $$ ℓ -th powers asymptotically. Based on these results we formulate problems concerning the maximum of $$P_{a,b;N}(\ell )$$ P a , b ; N ( ℓ ) , and we give affirmative answers to these questions for certain small values of $$\ell $$ ℓ and N.

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29

Lagrange, Jean, and John Leech. "Pythagorean ratios in arithmetic progression, part II. Four Pythagorean ratios." Glasgow Mathematical Journal 36, no. 1 (January 1994): 45–55. http://dx.doi.org/10.1017/s0017089500030536.

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As in [3] let {a, b}designate the Pythagorean ratio (a2 − b2)/2ab between the sides of a rational right angled triangle. The principal result of [3] is that {a, b}is the arithmetic mean of two Pythagorean ratios, and hence is the middle term of a three term arithmetic progression, if and only if a /b is the geometric mean of two Pythagorean ratios. Here in Part II we study sets of four Pythagorean ratios in arithmetic progression. We show that sets of four in consecutive places in an arithmetic progression are closely related to sets of four in the first, second, third and fifth places in a progression; any one of the former sets determines two of the latter sets, and either one of the latter sets determines the other and the former. We construct an infinite sequence of sets of four ratios in consecutive places of arithmetic progressions, the last term of each set being the first term of the next set. These sets are related to solutions of the Diophantine equations r2 = 5p2q2 ± 4(p4 − 2q4). Computer searches, in addition to exhibiting enough members of this sequence to enable us to identify it, also exhibited two sets which do not belong to this sequence.

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30

Ulas, Maciej. "Rational Points in Arithmetic Progressions on y2 = xn + k." Canadian Mathematical Bulletin 55, no. 1 (March 1, 2012): 193–207. http://dx.doi.org/10.4153/cmb-2011-058-1.

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AbstractLet C be a hyperelliptic curve given by the equation y2 = f(x) for f ∈ ℤ[x] without multiple roots. We say that points Pi = (xi, yi) ∈ C(ℚ) for i = 1, 2, … , m are in arithmetic progression if the numbers xi for i = 1, 2, … , m are in arithmetic progression.In this paper we show that there exists a polynomial k ∈ ℤ[t] with the property that on the elliptic curve ε′ : y2 = x3+k(t) (defined over the field ℚ(t)) we can find four points in arithmetic progression that are independent in the group of all ℚ(t)-rational points on the curve Ε′. In particular this result generalizes earlier results of Lee and Vélez. We also show that if n ∈ ℕ is odd, then there are infinitely many k's with the property that on curves y2 = xn + k there are four rational points in arithmetic progressions. In the case when n is even we can find infinitely many k's such that on curves y2 = xn +k there are six rational points in arithmetic progression.

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31

Koukoulopoulos, Dimitris. "Primes in short arithmetic progressions." International Journal of Number Theory 11, no. 05 (August 2015): 1499–521. http://dx.doi.org/10.1142/s1793042115400035.

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Let x, h and Q be three parameters. We show that, for most moduli q ≤ Q and for most positive real numbers y ≤ x, every reduced arithmetic progression a( mod q) has approximately the expected number of primes p from the interval (y, y + h], provided that h > x1/6+ϵ and Q satisfies appropriate bounds in terms of h and x. Moreover, we prove that, for most moduli q ≤ Q and for most positive real numbers y ≤ x, there is at least one prime p ∈ (y, y + h] lying in every reduced arithmetic progression a( mod q), provided that 1 ≤ Q2 ≤ h/x1/15+ϵ.

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32

Maltsev, Yu N., and A. S. Monastyreva. "On Triangles with Sides That Form an Arithmetic Progression." Izvestiya of Altai State University, no. 1(111) (March 6, 2020): 111–14. http://dx.doi.org/10.14258/izvasu(2020)1-18.

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Properties of triangles such that the squares of their sides form an arithmetic progression were studied in 2018. In this paper, triangles with sides that form an arithmetic progression are described. Let a, b, c be sides of an arbitrary triangle ABC. If sides b, a, c of the triangle ABC form an arithmetic progression then, for example, the equality a=(b+c)/2 (b<a<c) holds. The class of triangles for which a=(b+c)/2 is greater than the class of triangles for which b, a, c form an arithmetic progression. In this paper, we study the properties of triangles for which this equality holds. Thus, triangles with sides that form an arithmetic progression are described with the help of the parameters p, R, r. Classes of rectangular triangles, triangles with angle 30°, triangles with angle 60°, triangles with angle 120° are studied and described.

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33

Gupta, Soma, and Amitabha Tripathi. "Density of M-sets in arithmetic progression." Acta Arithmetica 89, no. 3 (1999): 255–57. http://dx.doi.org/10.4064/aa-89-3-255-257.

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34

Chisholm, C., and J. A. MacDougall. "Rational tetrahedra with edges in arithmetic progression." Journal of Number Theory 111, no. 1 (March 2005): 57–80. http://dx.doi.org/10.1016/j.jnt.2004.07.009.

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35

Xarles, Xavier. "Squares in arithmetic progression over number fields." Journal of Number Theory 132, no. 3 (March 2012): 379–89. http://dx.doi.org/10.1016/j.jnt.2011.07.010.

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36

Abu-Saymeh, Sadi, and Mowaffaq Hajja. "Triangles whose sides form an arithmetic progression." Mathematical Gazette 104, no. 561 (October 8, 2020): 469–81. http://dx.doi.org/10.1017/mag.2020.101.

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This article is motivated by, and is meant as a supplement to, the recent paper [1]. That paper proves three geometric characterisations of triangles whose sides are in arithmetic progression, or equivalently triangles in which one of the sides is the arithmetic mean of the other two. More precisely, it gives three geometric contexts in which such triangles appear. In this Article, we supply references for the results in [1] and we provide more proofs of these results. We also add more contexts in which such triangles appear, and we raise related issues for future work. We hope that this will be a source of problems for training for, and for including in, mathematical competitions.

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37

Brudern, J., and T. D. Wooley. "SPARSE VARIANCE FOR PRIMES IN ARITHMETIC PROGRESSION." Quarterly Journal of Mathematics 62, no. 2 (September 18, 2009): 289–305. http://dx.doi.org/10.1093/qmath/hap029.

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38

Wodzak, M. A. "Primes in arithmetic progression and uniform distribution." Proceedings of the American Mathematical Society 122, no. 1 (January 1, 1994): 313. http://dx.doi.org/10.1090/s0002-9939-1994-1233985-1.

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39

Baker, Roger, and Tristan Freiberg. "Sparser variance for primes in arithmetic progression." Monatshefte für Mathematik 187, no. 2 (November 15, 2017): 217–36. http://dx.doi.org/10.1007/s00605-017-1139-6.

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40

Shao, Zehui, Fei Deng, Meilian Liang, and Xiaodong Xu. "On sets without k-term arithmetic progression." Journal of Computer and System Sciences 78, no. 2 (March 2012): 610–18. http://dx.doi.org/10.1016/j.jcss.2011.09.003.

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41

TANNER, NOAM. "STRINGS OF CONSECUTIVE PRIMES IN FUNCTION FIELDS." International Journal of Number Theory 05, no. 01 (February 2009): 81–88. http://dx.doi.org/10.1142/s1793042109001918.

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In a recent paper, Thorne [5] proved the existence of arbitrarily long strings of consecutive primes in arithmetic progressions in the polynomial ring 𝔽q[t]. Here we extend this result to show that given any k there exists a string of k consecutive primes of degree D in arithmetic progression for all sufficiently large D.

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42

Győry, K., L. Hajdu, and N. Saradha. "On the Diophantine Equation n(n + d) · · · (n + (k − 1)d) = byl." Canadian Mathematical Bulletin 47, no. 3 (September 1, 2004): 373–88. http://dx.doi.org/10.4153/cmb-2004-037-1.

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AbstractWe show that the product of four or five consecutive positive terms in arithmetic progression can never be a perfect power whenever the initial term is coprime to the common difference of the arithmetic progression. This is a generalization of the results of Euler and Obláth for the case of squares, and an extension of a theorem of Győry on three terms in arithmetic progressions. Several other results concerning the integral solutions of the equation of the title are also obtained. We extend results of Sander on the rational solutions of the equation in n, y when b = d = 1. We show that there are only finitely many solutions in n, d, b, y when k ≥ 3, l ≥ 2 are fixed and k + l > 6.

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González-Jiménez, Enrique, and Xavier Xarles. "On a conjecture of Rudin on squares in arithmetic progressions." LMS Journal of Computation and Mathematics 17, no. 1 (2014): 58–76. http://dx.doi.org/10.1112/s1461157013000259.

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AbstractLet $Q(N;q,a)$ be the number of squares in the arithmetic progression $qn+a$, for $n=0$,$1,\ldots,N-1$, and let $Q(N)$ be the maximum of $Q(N;q,a)$ over all non-trivial arithmetic progressions $qn + a$. Rudin’s conjecture claims that $Q(N)=O(\sqrt{N})$, and in its stronger form that $Q(N)=Q(N;24,1)$ if $N\ge 6$. We prove the conjecture above for $6\le N\le 52$. We even prove that the arithmetic progression $24n+1$ is the only one, up to equivalence, that contains $Q(N)$ squares for the values of $N$ such that $Q(N)$ increases, for $7\le N\le 52$ ($N=8,13,16,23,27,36,41$ and $52$).Supplementary materials are available with this article.

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44

MATOMÄKI, KAISA. "CARMICHAEL NUMBERS IN ARITHMETIC PROGRESSIONS." Journal of the Australian Mathematical Society 94, no. 2 (March 8, 2013): 268–75. http://dx.doi.org/10.1017/s1446788712000547.

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AbstractWe prove that when $(a, m)= 1$ and $a$ is a quadratic residue $\hspace{0.167em} \mathrm{mod} \hspace{0.167em} m$, there are infinitely many Carmichael numbers in the arithmetic progression $a\hspace{0.167em} \mathrm{mod} \hspace{0.167em} m$. Indeed the number of them up to $x$ is at least ${x}^{1/ 5} $ when $x$ is large enough (depending on $m$).

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Jungic, V., J. Licht, M. Mahdian, J. Nesetril, and R. Radoicic. "Rainbow Arithmetic Progressions and Anti-Ramsey Results." Combinatorics, Probability and Computing 12, no. 5-6 (November 2003): 599–620. http://dx.doi.org/10.1017/s096354830300587x.

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The van der Waerden theorem in Ramsey theory states that, for every k and t and sufficiently large N, every k-colouring of [N] contains a monochromatic arithmetic progression of length t. Motivated by this result, Radoičić conjectured that every equinumerous 3-colouring of [3n] contains a 3-term rainbow arithmetic progression, i.e., an arithmetic progression whose terms are coloured with distinct colours. In this paper, we prove that every 3-colouring of the set of natural numbers for which each colour class has density more than 1/6, contains a 3-term rainbow arithmetic progression. We also prove similar results for colourings of . Finally, we give a general perspective on other anti-Ramsey-type problems that can be considered.

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SUN, XUE-GONG, and JIN-HUI FANG. "ON THE DENSITY OF INTEGERS OF THE FORM (p−1)2−n IN ARITHMETIC PROGRESSIONS." Bulletin of the Australian Mathematical Society 78, no. 3 (December 2008): 431–36. http://dx.doi.org/10.1017/s0004972708000804.

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AbstractErdős and Odlyzko proved that odd integers k such that k2n+1 is prime for some positive integer n have a positive lower density. In this paper, we characterize all arithmetic progressions in which natural numbers that can be expressed in the form (p−1)2−n (where p is a prime number) have a positive proportion. We also prove that an arithmetic progression consisting of odd numbers can be obtained from a covering system if and only if those integers in such a progression which can be expressed in the form (p−1)2−n have an asymptotic density of zero.

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47

Knaf, Hagen, Erich Selder, and Karlheinz Spindler. "Four rational squares in arithmetic progressions and a family of elliptic curves with positive Mordell-Weil rank." Mathematische Semesterberichte 67, no. 2 (June 30, 2020): 213–36. http://dx.doi.org/10.1007/s00591-020-00279-z.

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Abstract We study the question at which relative distances four squares of rational numbers can occur as terms in an arithmetic progression. This number-theoretical problem is seen to be equivalent to finding rational points on certain elliptic curves. Both number-theoretical results and results concerning the associated elliptic curves are derived; i.e., the correspondence between rational squares in arithmetic progressions and elliptic curves is exploited both ways.

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TSAI, MU-TSUN, and ALEXANDRU ZAHARESCU. "ON THE SUM OF CONSECUTIVE INTEGERS IN SEQUENCES II." International Journal of Number Theory 08, no. 05 (July 6, 2012): 1281–99. http://dx.doi.org/10.1142/s1793042112500753.

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Let A be a sequence of natural numbers, rA(n) be the number of ways to represent n as a sum of consecutive elements in A, and MA(x) ≔ ∑n ≤ x rA(n). We give a new short proof of LeVeque's formula regarding MA(x) when A is an arithmetic progression, and then extend the proof to give asymptotic formulas for the case when A behaves almost like an arithmetic progression, and also when A is the set of primes in an arithmetic progression.

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49

Wright, Steve. "Quadratic residues and non-residues in arithmetic progression." Journal of Number Theory 133, no. 7 (July 2013): 2398–430. http://dx.doi.org/10.1016/j.jnt.2013.01.004.

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50

Park, Young Ho, and June Bok Lee. "Permutation polynomials with exponents in an arithmetic progression." Bulletin of the Australian Mathematical Society 57, no. 2 (April 1998): 243–52. http://dx.doi.org/10.1017/s0004972700031622.

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Abstract:

We examine the permutation properties of the polynomials of the type hk, r, s(x) = xr (1 + xs + … + xsk) over the finite field , of characteristic p. We give sufficient and necessary conditions in terms of k and r for hk, r, l(x) to be a permutation polynomial over , for q = p or p2. We also prove that if hk, r, s(x) is a permutation polynomial over , then (k + 1)s = ±1.

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Journal articles on the topic 'Arithmetic Progression'

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