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Primitive in Arithmetic Progression

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Thao Dola
13/03/2017 at 13:18
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Let S = {\(\dfrac{1}{r}:\)r = 1,2,3,...}. For all integer k>1, prove that there is a k-term arithmetic progression in S such that no addition term in S can be added to it to form a (k+1)-term arithmetic progression.

Primitive in Arithmetic Progression


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Thao Dola
13/03/2017 at 13:20
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Prove that for every positive integers s,a,b with gcd(a,b) = 1, there are infinitely many integers n such that an + b  is product of s pairwise distinct prime numbers.

Primitive in Arithmetic Progression


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Thao Dola
13/03/2017 at 13:23
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Let \(p_k\) be the k-th prime number. For every integer N, prove that there exists a positive integer k such that both \(p_{k-1}\) and \(p_{\left(k+1\right)}\) are not in the interval [\(p_k-N\),\(p_k+N\)].

Primitive in Arithmetic Progression


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Thao Dola
13/03/2017 at 13:25
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Prove that if f(x) is pily nomial with rational coefficcients such that f(p) is a prime number for every prime number p, then either f(x) = x for all x or f(x) is the same prime constant for all x.

Primitive in Arithmetic Progression


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Thao Dola
13/03/2017 at 13:30
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Prove that for every pair of positive integers n and N, there are consecutive positive intefers k,k+1,...,k+N such that \(\varphi\left(k\right),\varphi\left(k+1\right),...,\varphi\left(k+N\right)\)are all divisible by n, where \(\varphi\left(n\right)\) is as defined in Euler's theorem.

Primitive in Arithmetic Progression


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Thao Dola
13/03/2017 at 13:07
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Des thte exist an inftinite squence of positive integers \(a_1,a_2,a_3,...\) such that \(a_m\) and \(a_n\) are copime if and only if |m-n| = 1?

Primitive in Arithmetic Progression


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games 18  double counting 8  generating functions 2  probabilistic method 1  Polynomial 9  inequality 13  area 17  Equation 9  Primitive Roots Modulo Primes 1  Primitive in Arithmetic Progression 6  Base n Representatioons 4  Pell Equation 1  mixed number 1  Fraction 29  Circle 3  Imaginary numbers 1  Decimal number 2  Volume 2  percentages 6  simple equation 19  absolute value 19  rational numbers 20  Operation of Indices 21  Simulataneous Equation A System of Equations 25  Multiplication of Polynomial 17  divisibility 24  Maximum 5  Minimum 8  Fraction 4  Prime Numbers and Composite Numbers 13  Square Number 26  Even and Odd Numbers 13  GDC and LCM 11  GCD and LCM 12  Permutation and combination 9  combinations 5  interger 7  number 10  Diophantine equations 2  equations 1  Grade 6 19  Power 3  equality 2  maxima-minima 2  square root 1  Polygon 2  IGCSE 1  factorial 1  integers 2 
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