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A Fast Algorithm for Computing Binomial Coefficients Modulo Powers of Two [PDF]
The Scientific World Journal, 2013 I present a new algorithm for computing binomial coefficients modulo . The proposed method has an preprocessing time, after which a binomial coefficient with can be computed modulo in time.
Mugurel Ionut Andreica
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On Prime Divisors of Binomial Coefficients [PDF]
Mathematics of Computation, 1988 This paper, using computational and theoretical methods, deals with prime divisors of binomial coefficients: Geometric distribution and number of distinct prime divisors are studied. We give a numerical result on a conjecture by Erdős on square divisors of binomial coefficients.
Pierre Goetgheluck
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Generating binomial coefficients in a row of Pascal's triangle from extensions of powers of eleven [PDF]
Heliyon, 2022 Sir Isaac Newton noticed that the values of the first five rows of Pascal's triangle are each formed by a power of 11, and claimed that subsequent rows can also be generated by a power of 11.
Md. Shariful Islam +3 more
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Binomial coefficients and Jacobi sums [PDF]
Transactions of the American Mathematical Society, 1984 Throughout this paper e e denotes an integer ⩾ 3 \geqslant 3 and p p a prime ≡ 1 ( mod e ) \equiv \;1\ \pmod e . With f f defined by p = e f
Richard H. Hudson, Kenneth S. Williams
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Number of Odd Binomial Coefficients [PDF]
Proceedings of the American Mathematical Society, 1977 Let F ( n ) F(n) denote the number of odd numbers in the first n rows of Pascal’s triangle, and θ = ( log 3 ) / log 2 ) \theta = (\log 3)/\log 2) . Then α = lim
Heiko Harborth
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Some congruence properties of binomial coefficients and linear second order recurrences [PDF]
International Journal of Mathematics and Mathematical Sciences, 1988 Using elementary methods, the following results are obtained:(I) If p is prime, 0≤m≤n ...
Neville Robbins
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Sum of the Reciprocals of the Binomial Coefficients
European Journal of Combinatorics, 1993 Let \(S_ n=\sum^ n_{k=0}{1 \over {n \choose k}}\). It is shown that \(S_ n\) satisfies the recurrence \(S_ n={n+1 \over 2n}S_{n-1}+1\). The proof can be simplified by observing that \[ n!S_ n=\sum^ n_{k=0}k!(n-k)!=n!+\sum^{n-1}_{k=0}k!(n-k-1)!(n+1-k-1) =n!+(n+ 1)(n-1)!S_{n-1}-(n!S_ n-n!). \]
B. Sury
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Binomial Coefficients and Quadratic Fields [PDF]
Proceedings of the American Mathematical Society, 2004 Let E be a real quadratic field with discriminant d and let p be an odd prime not dividing d.
Zhi‐Wei Sun
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Congruences for sums of binomial coefficients
Journal of Number Theory, 2007 Let q>1 and m>0 be relatively prime integers. We find an explicit period $ _m(q)$ such that for any integers n>0 and r we have $[n+ _m(q),r]_m(a)=[n,r]_m(a) (mod q)$ whenever a is an integer with $\gcd(1-(-a)^m,q)=1$, or a=-1 (mod q), or a=1 (mod q) and 2|m, where $[n,r]_m(a)=\sum_{k=r(mod m)}\binom{n}{k}a^k$.
Zhi‐Wei Sun, Roberto Tauraso
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Sums involving the binomial coefficients, Bernoulli numbers of the second kind and harmonic numbers [PDF]
Notes on Number Theory and Discrete Mathematics, 2023 We offer a number of various finite and infinite sum identities involving the binomial coefficients, Bernoulli numbers of the second kind and harmonic numbers. For example, among many others, we prove Σⁿₖ₌ₒ((-1)ᵏhₖ/4ᵏ)$binom{2k}{k}$Gₙ₋ₖ = ((-1)ⁿ⁻¹/2^²ⁿ⁻¹)
Necdet Batır, Anthony Sofo
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