Geodesic
On the equation $\varphi (|x^m-y^m|)=2^n$
Mathematica Bohemica, Tome 125 (2000) no. 4, pp. 465-479

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MR Zbl
In this paper we investigate the solutions of the equation in the title, where $\phi$ is the Euler function. We first show that it suffices to find the solutions of the above equation when $m=4$ and $x$ and $y$ are coprime positive integers. For this last equation, we show that aside from a few small solutions, all the others are in a one-to-one correspondence with the Fermat primes.
In this paper we investigate the solutions of the equation in the title, where $\phi$ is the Euler function. We first show that it suffices to find the solutions of the above equation when $m=4$ and $x$ and $y$ are coprime positive integers. For this last equation, we show that aside from a few small solutions, all the others are in a one-to-one correspondence with the Fermat primes.
DOI : 10.21136/MB.2000.126267
Classification : 11A25, 11A51, 11A63
Keywords: Euler function; Fermat primes 
Luca, Florian. On the equation $\varphi (|x^m-y^m|)=2^n$. Mathematica Bohemica, Tome 125 (2000) no. 4, pp. 465-479. doi: 10.21136/MB.2000.126267
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[1] R. D. Charmichael: On the numerical factors of arithmetic forms $\alpha^n ± \beta^n$. Ann. Math. 15 (1913-1914), 30-70.

[2] F. Luca: Equations involving arithmetic functions of Fibonacci and Lucas numbers. Preprint. To appear in Fibo. Quart. | MR | Zbl

[3] F. Luca: Pascal's triangle and constructible polygons. Preprint, | Zbl

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