## On the $β3x+1β$ problem

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- by R. E. Crandall PDF
- Math. Comp.
**32**(1978), 1281-1292 Request permission

## Abstract:

It is an open conjecture that for any positive odd integer*m*the function \[ C(m) = (3m + 1)/{2^{e(m)}},\] where $e(m)$ is chosen so that $C(m)$ is again an odd integer, satisfies ${C^h}(m) = 1$ for some

*h*. Here we show that the number of $m \leqslant x$ which satisfy the conjecture is at least ${x^c}$ for a positive constant

*c*. A connection between the validity of the conjecture and the diophantine equation ${2^x} - {3^y} = p$ is established. It is shown that if the conjecture fails due to an occurrence $m = {C^k}(m)$, then

*k*is greater than 17985. Finally, an analogous "$qx + r$" problem is settled for certain pairs $(q,r) \ne (3,1)$.

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## Additional Information

- © Copyright 1978 American Mathematical Society
- Journal: Math. Comp.
**32**(1978), 1281-1292 - MSC: Primary 10A99
- DOI: https://doi.org/10.1090/S0025-5718-1978-0480321-3
- MathSciNet review: 0480321