Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.
Updated
Sander 1992 prime power divisors binomial coefficients
theorem_1: States that for 0 < epsilon < 1 and every positive integer a, every binomial coefficient C(m,k) with m large and |m - 2k| < m^(1-epsilon) is divisible by p^a for some prime p > (1/2) m^(1/(a+1)).
theorem_2: States an upper bound for the sum over primes p <= N of e(x(h_1/p^(j_1) + ... + h_r/p^(j_r))) when 2 <= N <= x^(1/j), the paper's main tool for its other results.
theorem_3: States an asymptotic formula, with error term, for the number of primes p <= P such that the fractional part of x/p^j is below sigma_j for each 1 <= j <= J, valid for 2 <= P <= x^(1/J).
Sander, J. W., Prime power divisors of binomial coefficients. J. Reine Angew. Math. 430 (1992), 1--20. The file, served by the Hannover repository (repo.uni-hannover.de) and stamped "Bereitgestellt von Technische Informationsbibliothek Hannover", prints "© Walter de Gruyter Berlin · New York 1992" in the journal head of its first page, every other right reserved.
Sander answers all three questions raised by Erdős and by Erdős and Graham about high prime-power divisors of binomial coefficients. Theorem 1 (p. 1) states that for 0 < epsilon < 1 and a in N there is m_0 = m_0(epsilon, a) such that for all m >= m_0 and all 0 <= k <= m with |m - 2k| < m^{1-epsilon} (condition (1)), p^a divides C(m,k) for some prime p > (1/2) m^{1/(a+1)}; this simultaneously gives that middle binomial coefficients are never squarefree for large arguments (Erdős's 1975 conjecture, first settled for large n by Sárközy in 1985), that the a-th power divisor exists for every fixed a, and that the prime p may be taken to tend to infinity, and it covers shifted coefficients C(2n ± d, n) for d not too large. The main tool is Theorem 2 (p. 13), an upper bound for the exponential sum over primes sum_{p <= N} e(x(h_1/p^{j_1} + ... + h_r/p^{j_r})), generalizing estimates of Jutila and of the author, from which Theorem 3 (p. 14) deduces an asymptotic formula for the number of primes p <= P with {x/p^j} < sigma_j for 1 <= j <= J, where 0 < sigma_j <= 1. Section 2 sets up the preliminaries for the exponential sum with parameters r, real h_i and positive integers j_i, and the paper announces a sequel applying these estimates to Sárközy's method to obtain upper and lower bounds for the highest a-th power dividing binomial coefficients. For Problem 175 it gives, for all sufficiently large n only, a new proof that C(2n, n) is not squarefree (Sárközy's 1985 result), strengthened to divisibility by p^a for a prime p > (1/2) (2n)^{1/(a+1)}; it does not reach every n >= 5.
Source: https://repo.uni-hannover.de/handle/123456789/3179.
Bears on. #175: Theorem 1 (p. 1) with a = 2, m = 2n and k = n gives, for every sufficiently large n, a prime p > (1/2) (2n)^{1/3} with p^2 dividing C(2n, n), so C(2n, n) is not squarefree for all large n; the paper gives no explicit threshold and does not reach every n >= 5 (theorem_1).
Results. Page numbers are those of the journal print.
- Theorem 1 (p. 1): for 0 < epsilon < 1 and a in N there is m_0(epsilon, a) such that for all m >= m_0 and all 0 <= k <= m with |m - 2k| < m^{1-epsilon}, p^a divides C(m,k) for some prime p > (1/2) m^{1/(a+1)}.
- Theorem 2 (Section 3, p. 13; main tool): under the conventions (2) and (3) of p. 2, for 2 <= N <= x^{1/j}, the sum over primes p <= N of e(x(h_1/p^{j_1} + ... + h_r/p^{j_r})) is << (N^{1-c Lambda(N,xH)} + N^{(j+2)/2} x^{-1/2} + N^{5/6} H^2) (log xH)^{4J}.
- Theorem 3 (Section 4, p. 14): for 2 <= P <= x^{1/J} and 0 < sigma_j <= 1, the number of primes p <= P with {x/p^j} < sigma_j (1 <= j <= J) is sigma_1 ... sigma_J pi(P) + O(P^{1-c Lambda(P,x)} + P^{(J+2)/2+epsilon} x^{-1/2}) (log x)^{4J} for every epsilon > 0, where Lambda(X,Y) = (log X / log Y)^2 and the constants depend only on J.
Read status. Claims checked for Theorems 1, 2 and 3, read clause by clause on the print together with the conventions of Section 2 (p. 2); the proofs were read for their structure only.
No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.