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In this paper, we prove that positive integer solutions {a_n} to where the c's are nonnegative integers, and d = c_1 + c_2 +···+ c_k, have the property thateither {a_n} is periodic with period at most k, or {a_n} is unbounded.
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Fix positive integers m, n and let f be a real-valued function defined on an (arbitrary) given subset E ? R~n. How can we tell whether f extends to a C~m function F on the whole R~n? If such an F exists, then how small can we take...
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Fix positive integers m, n and let f be a real-valued function defined on an (arbitrary) given subset E ? R~n. How can we tell whether f extends to a C~m function F on the whole R~n? If such an F exists, then how small can we take its C~m-norm? What can we say about the derivatives ?~αF(x) at a given point x? Can we take F to depend linearly on f?
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An open question going back to Euler (see Dickson [4]) is the solution in positive integers of We show solutions in integers w, x on the left side and Gaussian integers y, z on right side. We also give an identity solution where w...
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An open question going back to Euler (see Dickson [4]) is the solution in positive integers of We show solutions in integers w, x on the left side and Gaussian integers y, z on right side. We also give an identity solution where w, x, y, and z are all Gaussian integers. The solutions to for n = 4 are well known and date back to Euler. For integers n ≥ 5 it's an open question that solutions even exist, let alone finding any. See Hardy and Wright [5] and the ebook by Piezas [6], for example. The case where n = 3 involves the well known "Taxicab numbers" named after the famous Hardy and Ramanujan hospital story. (See Hardy [3].) In our present note we have drawn upon the 2015 preprint by Campbell and Zujev [2] and unpublished work from the author.
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For every integer c, let n = R_d(c) be the least integer such that for every coloring △: {1,2,...,n}→{0,1} there exists a solution (x_1,x_2,x_3) to x_1 + x_2 + C = x_3 such that x_i≠x_j when i≠j and △(x_1) = △(x_2) = △(x_3)...
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For every integer c, let n = R_d(c) be the least integer such that for every coloring △: {1,2,...,n}→{0,1} there exists a solution (x_1,x_2,x_3) to x_1 + x_2 + C = x_3 such that x_i≠x_j when i≠j and △(x_1) = △(x_2) = △(x_3).In this paper it is shown that for every integer c, R_d(c={4c+8 if c≥18 if -3≤c≤-69 if c=0,-2, -7,-810 if c=-1,-9|c|-└(|c|-4)/5 ┘ if c≤-10. Note: Part of the research for this paper occurred when the second author was an undergraduate student at South Dakota State University.
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For a given positive integer n, we determine explicit formulas for the numberof occurrences of n as a part of a Pythagorean triple, and also as a part of a primitivePythagorean triple. We also determine the least positive integer ...
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For a given positive integer n, we determine explicit formulas for the numberof occurrences of n as a part of a Pythagorean triple, and also as a part of a primitivePythagorean triple. We also determine the least positive integer that is a part of at least nsuch primitive triples and obtain several conditions that help in characterizing the analogouscase for all triples.
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A finite set A of integers is said to tile the set of integers Z if and only if the set of integers can be written as a disjoint union of translates of the set A. Newman [N] obtained necessary and sufficient condition for a set A ...
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A finite set A of integers is said to tile the set of integers Z if and only if the set of integers can be written as a disjoint union of translates of the set A. Newman [N] obtained necessary and sufficient condition for a set A to tile integers when |A| =p~a, p prime, a a positive integer. For a = 1, p = 3, he remarked that there should be a trivial proof. In this paper, a proof based on elementary number theory has been obtained for a = 1, p = 3.
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Reversible integer-to-integer (I2I) mapping of orthonormal transforms are vital for developing lossless coding with scalable decoding functionalities. A general framework for reversible I2I mapping of N-point, where N is a positiv...
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Reversible integer-to-integer (I2I) mapping of orthonormal transforms are vital for developing lossless coding with scalable decoding functionalities. A general framework for reversible I2I mapping of N-point, where N is a positive integer power of 2, orthonormal block transforms using recursive factorization of such transform matrices and the lifting scheme is presented. Designs include the discrete cosine transform (DCT) that maps integers to integers (I2I-DCT), the discrete sine transform that maps integers to integers (I2I-DST) and the Walsh-Hadamard transform that maps integer to integers (I2I-WHT). The main significant feature of these designs is that the transform coefficients are normalized according to the conventional scaling factors, which is vital for embedded coding, while preserving the integer-to-integer mapping and perfect reconstruction. This makes these transforms usable in both lossless and lossy image coding, especially in scalable lossless coding. These generic N-point design of the above transforms enables evaluating the effect of block sizes of such transforms in lossless coding. The performance is evaluated in terms of lossless image and video coding, quality scalable decoding, complexity and lifting step rounding effects.
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Text: Let the prime factorization of n be n = _(q1q2) ? _(q a) with _(q1) ≥ _(q2) ≥ ? ≥ _(q a) ≥ 2. A positive integer n is said to be ordinary if the smallest positive integer with exactly n divisors is p1q1-1p2q2-1?paqa-1, w...
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Text: Let the prime factorization of n be n = _(q1q2) ? _(q a) with _(q1) ≥ _(q2) ≥ ? ≥ _(q a) ≥ 2. A positive integer n is said to be ordinary if the smallest positive integer with exactly n divisors is p1q1-1p2q2-1?paqa-1, where _(p k) denotes the kth prime. In this paper I prove that all integers of the form ql are ordinary, where l is a square-free positive integer and q is a prime. This confirms a conjecture of Yong-Gao Chen. Video: For a video summary of this paper, please click here or visit http://youtu.be/WTY4wr8L_U0.
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In spite of the provable rarity of integer matrices with integer eigenvalues, they are commonly used as examples in introductory courses. We present a quick method for constructing such matrices starting with a given set of eigenv...
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In spite of the provable rarity of integer matrices with integer eigenvalues, they are commonly used as examples in introductory courses. We present a quick method for constructing such matrices starting with a given set of eigenvectors. The main feature of the method is an added level of flexibility in the choice of allowable eigenvalues. The method is also applicable to nondiagonalizable matrices, when given a basis of generalized eigenvectors. We have produced an online web tool that implements these constructions.
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We obtain upper bounds for the number of arbitrary and symmetric matrices with integer entries in a given box (in an arbitrary location) and a given determinant. We then apply these bounds to estimate the number of matrices in suc...
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We obtain upper bounds for the number of arbitrary and symmetric matrices with integer entries in a given box (in an arbitrary location) and a given determinant. We then apply these bounds to estimate the number of matrices in such boxes which have an integer eigenvalues. Finally, we outline some open questions
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