Ecklundjr 1969 prime divisors binomial coefficient

../

evidence/: Retains the independent full-proof review of the five components and the final exact-byte receipt.

external_inputs: Records the exact Sylvester--Schur, Rosser--Schoenfeld, and Faulkner inputs quoted or invoked in Ecklund's proof, with their hypotheses and uses.

lemma_1: Bounds a binomial coefficient with no prime divisor at most n over two by the primes in its numerator interval.

lemma_2: Uses the cited Rosser--Schoenfeld estimates to bound the prime product in an interval of length k when k is at least 59.

lemma_3: Gives Ecklund's inductive lower bound for a binomial coefficient whose top argument is a power of two times its lower argument.

theorem: Reconstructs Ecklund's complete proof that n choose k for n at least twice k has a prime divisor at most the larger of n over k and n over two, apart from 7 choose 3.

---

E. F. Ecklund, Jr., On prime divisors of the binomial coefficient, Pacific Journal of Mathematics 29 (1969), 267--270. The manuscript was received July 8, 1968; the issue is dated June 1969. No copyright line is printed in the publisher PDF (its cover page, pp. 267--270, the journal's back matter and the issue contents); the journal's article page shows "© Copyright 1969 Pacific Journal of Mathematics. All rights reserved." (https://msp.org/pjm/1969/29-2/p04.xhtml), every other right reserved.

The displayed theorem states that if $n\geq2k$, then $\binom nk$ has a prime divisor

$$p\leq\max\{n/k,n/2\},$$

with the exception $\binom73=35$. The maximum is essential at $k=1$; for example, $\binom51=5$ has no prime divisor at most $5/2$.

For $k\geq2$ within the theorem's range, the maximum is $n/2$. Symmetry therefore gives the corrected Problem 384 consequence: whenever $1<k<n-1$, $\binom nk$ has a prime divisor $p\leq n/2$, except for the coefficient $\binom73=\binom74$. The imported strict variant is false at $\binom62=15$.

The complete same-paper chain transcribed here consists of:

• [[factorials_binomials/ecklundjr_1969_prime_divisors_binomial_coefficient/lemma_1|Lemma 1]], which bounds a hypothetical counterexample by primes in $(n-k,n]$;
• [[factorials_binomials/ecklundjr_1969_prime_divisors_binomial_coefficient/lemma_2|Lemma 2]], which applies two exact Rosser--Schoenfeld estimates;
• [[factorials_binomials/ecklundjr_1969_prime_divisors_binomial_coefficient/lemma_3|Lemma 3]], the dyadic binomial lower bound;
• the theorem's three analytic cases, explicit small cases, complete range assembly, and exact replay of the two finite ranges; and
• [[factorials_binomials/ecklundjr_1969_prime_divisors_binomial_coefficient/external_inputs|the bounded external inputs]], including their hypotheses and applications.

Printed p.269 first derives

$$\frac{2^{4k-1}}{\sqrt{k}} <e^{k+2.06\sqrt{15k}},$$

then prints $2.6$ in the next logarithmic line while claiming the $k\geq25$ cutoff. That change is consequential. The reconstruction preserves the defect and uses the valid preceding $2.06$ display together with a project-authored exact threshold certificate. The theorem page's living verification record includes that repair, the remaining cases, and the full chain in its current accepted scope.

The copy read for this card is the publisher PDF, seven physical pages. The article occupies physical pp.2--5, corresponding to printed pp.267--270; physical p.1 is the article cover, p.6 journal back matter, and p.7 the issue contents. All seven pages were visually inspected. Native extraction was used only for navigation.

Source: https://msp.org/pjm/1969/29-2/p04.xhtml.

Bears on.

• Problem 384: by symmetry, the theorem gives every $\binom nk$ with $1<k<n-1$ a prime divisor $p\leq n/2$, except $\binom73=\binom74$; this is the problem's statement with the non-strict bound. It gives nothing toward the strict bound $p<n/2$, which fails at $\binom42$ and $\binom62$. Lemmas 1--3 bear on the problem only as steps of the theorem's proof.

Results transcribed.

• Theorem, printed pp.267--270: the complementary prime-divisor bound, exception, and complete proof.
• Lemma 1 and equation (6), printed pp.267--268.
• Lemma 2 and equation (7), printed pp.267--268.
• Lemma 3 and equation (8), printed p.268.
• Rosser--Schoenfeld equations (1)--(5), the contextual Sylvester--Schur theorem, and the Faulkner bound used in Case 2.
• Problem 384 consequence by symmetry. This transfer is a compilation consequence rather than a separately printed theorem.

Verification record

The living record is maintained on the [[factorials_binomials/ecklundjr_1969_prime_divisors_binomial_coefficient/theorem|theorem page]]. Its current state is accepted for five complete natural-language components: three same-paper lemmas, the main theorem chain, and the E384 symmetry transfer. The source version reviewed is the seven-page publisher PDF of the Pacific Journal of Mathematics 29 (1969), 267--270. The proof remains relative to the five quoted Rosser--Schoenfeld estimates and the Faulkner implication. Sylvester--Schur is contextual only. The external proofs have not been recursively reviewed. No formal verification is recorded.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.