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On Furstenberg’s proof of the infinitude of primes
 Amer. Math. Monthly
, 2009
"... Theorem. There are infinitely many primes. Euclid’s proof of this theorem is a classic piece of mathematics. And although one proof is enough to establish the truth of the theorem, many generations of mathematicians have amused themselves by coming up with alternative proofs. See, for example, [2], ..."
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the “real reason ” that Furstenberg’s approach works. Definition. If m and r are integers with m ≥ 1, we let r + mZ denote the set of integers congruent to r mod m, so for example, 2 + 7Z = 9 + 7Z = −5 + 7Z = {...,−12,−5, 2, 9, 16,...}. We call any such set an arithmetic progression, or AP for short
THE NONEXTENSIBILITY OF D(4k)TRIPLES f1; 4k(k 1); 4k2 + 1g WITH jkj PRIME
"... Abstract. For a nonzero integer n, a set of m distinct positive integers fa1; : : : ; amg is called a D(n)mtuple if aiaj + n is a perfect square for each i; j with 1 i < j m. Let k be an integer with jkj prime. Then we show that the D(4k)triple f1; 4k(k 1); 4k2 + 1g cannot be extended to a ..."
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Abstract. For a nonzero integer n, a set of m distinct positive integers fa1; : : : ; amg is called a D(n)mtuple if aiaj + n is a perfect square for each i; j with 1 i < j m. Let k be an integer with jkj prime. Then we show that the D(4k)triple f1; 4k(k 1); 4k2 + 1g cannot be extended
Experience in Factoring Large Integers Using Quadratic Sieve
, 2005
"... GQS is a set of computer programs for factoring “large ” integers. It is based on multiple polynomial quadratic sieve. The current version, 3.0, can factor a 82decimaldigit integer in a PC with AMD 1.8G Hz processor and 512 MB main memory in one day. The largest number I have factored using GQS i ..."
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GQS is a set of computer programs for factoring “large ” integers. It is based on multiple polynomial quadratic sieve. The current version, 3.0, can factor a 82decimaldigit integer in a PC with AMD 1.8G Hz processor and 512 MB main memory in one day. The largest number I have factored using GQS
Consistency for partition regular equations
 Discrete Math
"... It is easy to deduce from Ramsey’s Theorem that, given positive integers a1, a2,..., am and a finite colouring of the set N of positive integers, there exists an injective sequence (xi) ∞ i=1 with all sums of the form � m i=1 aixr i (r1 < r2 < · · · < rm) lying in the same colour class. ..."
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It is easy to deduce from Ramsey’s Theorem that, given positive integers a1, a2,..., am and a finite colouring of the set N of positive integers, there exists an injective sequence (xi) ∞ i=1 with all sums of the form � m i=1 aixr i (r1 < r2 < · · · < rm) lying in the same colour class
Journal
"... identify the ink colors of color words (for a review, see MacLeod, 1991). Responses are typically slower on incongruent trials (red in blue ink) than on congruent trials (blue in blue ink). The difference in response latency between congruent and incongruent trials is often taken as a measure of t ..."
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identify the ink colors of color words (for a review, see MacLeod, 1991). Responses are typically slower on incongruent trials (red in blue ink) than on congruent trials (blue in blue ink). The difference in response latency between congruent and incongruent trials is often taken as a measure
SpaceTime Transmission using TomlinsonHarashima Precoding
, 2002
"... In this paper, TomlinsonHarashima precoding, a nonlinear preequalization technique, is proposed for transmission over multipleinput/multipleoutput channels. Instead of equalizing intersymbol interference (temporal equalization) here spatial equalization, i.e., equalization of the multiuser inte ..."
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Cited by 45 (15 self)
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In this paper, TomlinsonHarashima precoding, a nonlinear preequalization technique, is proposed for transmission over multipleinput/multipleoutput channels. Instead of equalizing intersymbol interference (temporal equalization) here spatial equalization, i.e., equalization of the multiuser interference, or combined spatial/temporal equalization is performed. It is shown that this MIMe precodinglike its sISe counterpart offers significant advantages over linear preequalization and over decisionfeedback equalization, as is done in BLASTlike schemes. Using channel coding, MIMe precoding is able to achieve higher power efficiencies at lower coding delays than competing schemes.
Adult age differences in dual information processes: Implications for the role of affective and deliberative processes in older adults’ decision making
 Perspectives on Psychological Science
"... ABSTRACT—Age differences in affective/experiential and deliberative processes have important theoretical implications for judgment and decision theory and important pragmatic implications for olderadult decision making. Agerelated declines in the efficiency of deliberative processes predict poor ..."
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ABSTRACT—Age differences in affective/experiential and deliberative processes have important theoretical implications for judgment and decision theory and important pragmatic implications for olderadult decision making. Agerelated declines in the efficiency of deliberative processes predict poorerquality decisions as we age. However, agerelated adaptive processes, including motivated selectivity in the use of deliberative capacity, an increased focus on emotional goals, and greater experience, predict better or worse decisions for older adults depending on the situation. The aim of the current review is to examine adult age differences in affective and deliberative information processes in order to understand their potential impact on judgments and decisions. We review evidence for the role of these dual processes in judgment and decision making
'3 QUADRATIC SUBFIELDS OF THE SPLITTING FIELD OF A DIHEDRAL QUINTIC TRINOMIAL x5+ax+b
, 1995
"... Abstract. It is known that every quadratic field K is a subfield of the splitting field of a dihedral quintic polynomial. In this paper it is shown that K is a subfield of the splitting field of a dihedral quintic trinomial x5 + ax + b if and only if the discriminant of K is of the form49 or89, wh ..."
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, where q is the (possibly empty) product of distinct primes congruent to 1 modulo 4.
Classical 6j{symbols and the tetrahedron
, 1999
"... A classical 6j{symbol is a real number which can be associated to a labelling of the six edges of a tetrahedron by irreducible representations of SU(2). This abstract association is traditionally used simply to express the symmetry of the 6j{symbol, which is a purely algebraic object; however, it ha ..."
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are these dimensions. The goal of this paper is to prove and explain this formula by using geometric quantization. A surprising spino is that a generic Euclidean tetrahedron gives rise to a family of twelve scissorscongruent but noncongruent tetrahedra. AMS Classication numbers Primary: 22E99 Secondary: 81R05, 51M
Results 1  10
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