### Switchback Swagger III

An intriguing paper [1] by Kalman Gyory discusses the equation

$m(m+1) \cdots (m + i - 1) = b k^{l}$

For $b = 1$ Erdös and Selfridge proved in 1975 [2] that this equation has no non-trivial solutions in the positive integers. The $(i,l,b) = (3,2,24)$ case can be seen to correspond to the cannonball problem under the substitution $n \mapsto \frac{m}{2}$. In general this suggests that the sequence of switchback expressions

$P_i \frac{\textrm{sum of squares}}{in + T_i}$

may hardly ever be expressed in the form $b k^{l}$ for $k \geq 2$, where $T_i$ is the triangular number $\sum_{j=1}^{i} {j}$, even though it is certainly a positive integer. This is an interesting fact about the cardinality of these faces of the permutohedra, and for some mysterious reason the proof for $b=1$ seems to involve the mathematics of Fermat's last theorem. Note also the similarity between the denominator above and terms in the associahedra sequences $F_{n}(i)$.

[1] K. Gyory, Acta Arith. 83 (1998) 87-92

[2] P. Erdös and J.L. Selfridge, Illinois J. Math. 19 (1975) 292-301

$m(m+1) \cdots (m + i - 1) = b k^{l}$

For $b = 1$ Erdös and Selfridge proved in 1975 [2] that this equation has no non-trivial solutions in the positive integers. The $(i,l,b) = (3,2,24)$ case can be seen to correspond to the cannonball problem under the substitution $n \mapsto \frac{m}{2}$. In general this suggests that the sequence of switchback expressions

$P_i \frac{\textrm{sum of squares}}{in + T_i}$

may hardly ever be expressed in the form $b k^{l}$ for $k \geq 2$, where $T_i$ is the triangular number $\sum_{j=1}^{i} {j}$, even though it is certainly a positive integer. This is an interesting fact about the cardinality of these faces of the permutohedra, and for some mysterious reason the proof for $b=1$ seems to involve the mathematics of Fermat's last theorem. Note also the similarity between the denominator above and terms in the associahedra sequences $F_{n}(i)$.

[1] K. Gyory, Acta Arith. 83 (1998) 87-92

[2] P. Erdös and J.L. Selfridge, Illinois J. Math. 19 (1975) 292-301

## 2 Comments:

Congratulations on receiving your PhD! No one could deserve it more!

Thanks! It was fun: lots of good company, a parade through town (which is named after Christ Church College in Oxford), music, a speech by the Governor General of New Zealand, food, champagne etc. I'm hoping some people will send me photos to post.

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