Charles Babbage's Conjecture/Historical Note

Historical Note on Charles Babbage's Conjecture

The name Charles Babbage's Conjecture has been coined by $\mathsf{Pr} \infty \mathsf{fWiki}$ as a convenient label to apply to a statement whose full description is unwieldy.

Note that the name Babbage's Conjecture is already used in the literature to refer to a conjecture suggested by Dennis William Babbage.

According to David Wells's $1997$ work Curious and Interesting Numbers, 2nd ed., the refutation is the work of David Singmaster, who also determined that no other such counterexample exists for any $n$ less than $150 \, 000$.

However, this still needs to be corroborated by evidence on the internet.

Wells also suggests that any higher power of $16 \, 843$ is also a counterexample, which would be interesting, as it would suggest that, for example:

$16 \, 843^4 \divides \dbinom {2 \times 16 \, 843^2 - 1} {16 \, 843^2 - 1} - 1$

It appears prudent, therefore, to ascertain whether Wells is actually correct in his citation of $16 \, 843$ being the counterexample that Singmaster actually discovered.

Hence research is invited into the literature, to find the original work and to confirm what is claimed here.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $16,843$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $16,843$