A Note on Nested Sums
... Looking at the proof of Lemma 1 we see that the two main ingredients were the identities which allowed us to write a sum of binomial coefficients as a single binomial coefficient and rewrite a product of two binomial coefficients so that we could pull one term out. When our function is no longer a s ...
... Looking at the proof of Lemma 1 we see that the two main ingredients were the identities which allowed us to write a sum of binomial coefficients as a single binomial coefficient and rewrite a product of two binomial coefficients so that we could pull one term out. When our function is no longer a s ...
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... Theorem 2.2: Let {an} be a bounded sequence of integers with the property that an ^ 0 for infinitely many n. Suppose further that {Un} is a generalized Fibonacci sequence generated with respect to the relatively prime pair (P, Q), where | P | > 1 and \Q\ = 1. If I(n) = UkUk+l...Uk+f^, where k e N\ { ...
... Theorem 2.2: Let {an} be a bounded sequence of integers with the property that an ^ 0 for infinitely many n. Suppose further that {Un} is a generalized Fibonacci sequence generated with respect to the relatively prime pair (P, Q), where | P | > 1 and \Q\ = 1. If I(n) = UkUk+l...Uk+f^, where k e N\ { ...
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... In [3, p. 52], Richard Guy gives the following problem of Schinzel: If p is an odd prime and n = 2 or p or 2p, then (cj)(n) + 1) \n, where is Euler's
totient function. Is this true for any other n?
We shall show that this question is closely related to a much older problem
due to Lehmer [4]: whe ...
... In [3, p. 52], Richard Guy gives the following problem of Schinzel: If p is an odd prime and n = 2 or p or 2p, then (cj)(n) + 1) \n, where
ON FINITE SUMS OF RECIPROCALS OF DISTINCT
... interval [TΓ, π + σ) which is disjoint from any other interval [TΓ', TΓ' + σ) for TΓ Φ τr'eP r _!. Therefore Ac(S) = \Jnepr-.x\.π9π + σ) is the disjoint union of exactly 2r~1 half-open intervals [TΓ, TΓ + σ), πe Pr-19 (since there are exactly 2r~~1 formally distinct sums of the form Σί=i ε Λ>εk — 0 ...
... interval [TΓ, π + σ) which is disjoint from any other interval [TΓ', TΓ' + σ) for TΓ Φ τr'eP r _!. Therefore Ac(S) = \Jnepr-.x\.π9π + σ) is the disjoint union of exactly 2r~1 half-open intervals [TΓ, TΓ + σ), πe Pr-19 (since there are exactly 2r~~1 formally distinct sums of the form Σί=i ε Λ>εk — 0 ...
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... A rational number r is said to be divisible by a prime number p provided the numerator of r is divisible by p . Here it is assumed that all rational numbers are written in standard form. That is, the numerators and denominators are relatively prime integers and the denominators are positive. Certain ...
... A rational number r is said to be divisible by a prime number p provided the numerator of r is divisible by p . Here it is assumed that all rational numbers are written in standard form. That is, the numerators and denominators are relatively prime integers and the denominators are positive. Certain ...
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... mutually prime.) If his paper had been written by a much less eminent mathematician, I might have suspected that his claims were based in part on numerical evidence and not on complete proofs. The basic idea in Jacobifs proof is to note that much of ordinary number theory can be generalized to the G ...
... mutually prime.) If his paper had been written by a much less eminent mathematician, I might have suspected that his claims were based in part on numerical evidence and not on complete proofs. The basic idea in Jacobifs proof is to note that much of ordinary number theory can be generalized to the G ...
More about Permutations and Symmetry Groups
... previously, but this can get a little tedious if we have to write a lot of them (and we will). The most efficient notation is cycle notation, which we will explain. A permutation p is called a cycle of length k, or a k-cycle, if there exists a subset {a1 , a2 , . . . , ak } ✓ {1, 2, . . . , n} such ...
... previously, but this can get a little tedious if we have to write a lot of them (and we will). The most efficient notation is cycle notation, which we will explain. A permutation p is called a cycle of length k, or a k-cycle, if there exists a subset {a1 , a2 , . . . , ak } ✓ {1, 2, . . . , n} such ...
Group action
... Factorization of a + bi is a unit times some factors from the right. Norms of a + bi and a – bi are the same, so they should get the same number of factors. Each of them should get 11 (since they are conjugate) and two from the following 10 + i, 10 + i, 10 – i, 10 – i. That leaves us with only two o ...
... Factorization of a + bi is a unit times some factors from the right. Norms of a + bi and a – bi are the same, so they should get the same number of factors. Each of them should get 11 (since they are conjugate) and two from the following 10 + i, 10 + i, 10 – i, 10 – i. That leaves us with only two o ...
1994
... 19. (a) If x < 1/3 then y = -x - 4; if 1/3 < x < 1/2 then y = 5x - 6 and if x > 1/2 then y = x + 4. Thus y decreases to the left of x = 1/3 and increases to the right of x = 1/3; the minimum is at x = 1/3. 20. (e) Let r,s be the respective ratios of the a,b sequences. Then a1b1s = 1, a1b1r = 4/3, a1 ...
... 19. (a) If x < 1/3 then y = -x - 4; if 1/3 < x < 1/2 then y = 5x - 6 and if x > 1/2 then y = x + 4. Thus y decreases to the left of x = 1/3 and increases to the right of x = 1/3; the minimum is at x = 1/3. 20. (e) Let r,s be the respective ratios of the a,b sequences. Then a1b1s = 1, a1b1r = 4/3, a1 ...
