# Squarefree binomial coefficients.

At $$n=23$$, all binomial coefficients are squarefree. Is this the largest value for $$n$$ where this is the case?

# Edit

A plot up to $$n=50$$:

A plot up to $$n=500$$:

plotted against $$n+1$$ and $$\frac{112}{\sqrt{239}}\sqrt{n}$$

and the same up to $$n=2000$$:

Do these bounds hold for all $$n$$?

(Clearly, the bound $$n+1$$ holds for all $$n$$.)

# Update

Just out of interest, a plot up to $$n=2000$$ with bounding curve $$24\log(n)$$

($$24\approx\frac{148}{\log 479}$$, where $$148$$ is the number of squarefree binomial coefficients at $$n=479$$),

which seems to be the tightest curve which still holds for up to $$n=3967$$:

... seems to suggest that the bound is $$const. \log(n)$$, which so far shows $$c\approx24$$ (which, coincidentally, is the number of squarefree coefficients at $$n=23$$).

• I checked numerically up to $n=10^5$, and $23$ is still the largest. – Jean-Claude Arbaut Nov 28 '13 at 16:40
• How did you check up till there? Did you use a program? – Ahaan S. Rungta Nov 28 '13 at 16:41
• Yes. In GAP: Filtered([1..100000], n -> ForAll([0..n], k -> MoebiusMu(Binomial(n,k))<>0)); – Jean-Claude Arbaut Nov 28 '13 at 16:41
• Actually, it's the last, see OEIS sequence A048278 – Jean-Claude Arbaut Nov 28 '13 at 16:43
• In case one is interested, the next "record" value after n=479 (148 binomials are square free) is for n=3967 (182 square free). Checked up to 4500. – Jean-Claude Arbaut Nov 28 '13 at 22:11

## 1 Answer

See here. The $n=23$ case is the last.

• @ Ahaan Rungta - many thanks for your link :) – martin Nov 28 '13 at 16:59
• Actually, it's arbautjc's credit. See his comment. – Ahaan S. Rungta Nov 28 '13 at 17:01