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First we prove another case in which n satisfies the Condition 1 defined in the section "An open problem". Let q_{i} be the largest prime smaller than n and let p_{q}^{aq}
be any prime factor divisor of n. By checking computationally all the pairs of primes that satisfy Condition 1 for each integer, we prove the following: Theorem. If n-q_{i} < p_{q}^{aq}
, then n satisfies Condition 1 with p_{q} and q_{i}. For the proof of this theorem we make use of Bertrand-Chebyshev Theorem: Theorem. (Bertrand-Chebyshev) For every integer n > 3 there exists a prime p such that n/2 < p < n. Proof: For the proof of the previous theorem we distinguish between two intervals: the interval (1, n-q_{i}] and the interval (n-q_{i}, n]. Due to the symmetry of binomial coefficients, we only consider k £ n/2. By the Bertrand-Chebyshev Theorem, we know that there is at least one prime between n/2 and n, hence n/2 < q_{i} < n. Then, for all k, k < n/2 < q_{i}. The base q_{i} representation of n is 1 · q_{i} + n-q_{i}. Therefore, we do not need to consider the interval (n-q_{i}) because the last digit of the base q_{i} representation of any k> n-q_{i} is larger than the last digit of the base q_{i} representation of n and thus by Lucas¢ Theorem the binomial coefficient \binomnk is divisible by q_{i}. If also there is no multiple of p_{q}^{aq}
in the interval (1, n-q_{i}), then by Lucas¢ Theorem all the binomial coefficients \binomnk with 1 £ k £ n/1 are divisible by at least p_{q} or p_{i}. Moreover, the case of equality in the previous theorem cannot occur because p_{q}^{aq}
divides both p_{q}^{aq}
and n, and hence q_{i} would not be a prime.
LaTeX4Web 1.4 OUTPUT
Note that for the previous theorem we do not necessarily need p_{q}^{aq}
to be the largest prime power divisor of n, and therefore we have the following corollary: Corollary. Let p_{j}^{aj}
denote the largest prime power divisor of an integer n. If n - q_{i} < p_{j}^{aj}
, then n satisfies Condition 1 with p_{j} and q_{i}. Note that if n satisfies Condition 1 at least one of these two primes has to be a prime factor of n, because otherwise C(n,1) = n is not divisible by either of the two primes. Thus, the only remaining cases are those in which n - q_{i} > p_{q}^{aq}
and n is not a prime or a prime power. By analyzing the integers that are part of these remaining cases, we notice that n usually satisfies Condition 1 with the pair formed by a prime factor of n and the largest prime smaller than n/2. Let q_{r} denote the largest prime smaller than n/2. If we analyze the six numbers smaller than 2,000 such that n - q_{i} > p_{q}^{aq}
, we see that the inequality p_{q}^{aq}
> n-2q_{r} holds and q_{r} and p_{q} satisfy Condition 1. The following table shows the only six numbers until 2,000 that do not satisfy Condition 1 with q_{i} and p_{q}. However, the sequence of all such integers is infinite. The Online Encyclopedia of Integer Sequences (OEIS) has accepted our submission of this sequence with the reference A290203. Check the webpage: https://oeis.org/A290203.
LaTeX4Web 1.4 OUTPUT
Then, we can study the inequality n-q_{i} > p_{q}^{aq}
> n-2q_{r}. The fact that we are considering n-2q_{r} comes from the base q_{r} representation of n. We also computed the sequence of numbers that do not satisfy this inequality, and the OEIS has accepted our sequence with the reference A290290. Check the webpage: https://oeis.org/A290290.
Here you can find the code for computing the sequences A290203 and A290290 respectively.
LaTeX4Web 1.4 OUTPUT
In general, for any natural integer d, we can study the function n-dq_{d}, where q_{d} refers to the closest prime to n/d.
We consider the integers n that do not satisfy the inequality p_{q}^{aq}
> n-2p_{2}. Up to 1,000,000 there are only 88 integers that do not satisfy p_{q}^{aq}
> n-2p_{2}. Up to 1,000,000 there are 25 integers that do not satisfy the inequality p_{q}^{aq}
> n-3p_{3}. Until 1,000,000, there are 7 integers that do not satisfy the inequality p_{q}^{aq}
> n-4p_{4}, 5 integers that do not satisfy the inequality p_{q}^{aq}
> n-5p_{5} and only 1 integer that does not satisfy the inequality p_{q}^{aq}
> n-6p_{6}. The following figure shows the number of integers up to 1,000,000 that do not satisfy the inequality p_{q}^{aq}
depending on d.