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Doess the above curve have only two rational points namely $(x,y)=(0,1)$ and $(-1,1)$ ?

Q_p
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    Please provide more context and show some effort (what have you tried so far? What experience do you have in this subject?) – Brevan Ellefsen Dec 27 '15 at 22:16
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    As per Faltings theorem there are only finitely many rational points. I don't know if there is an efficient method for finding all of them. The tools are likely very specialized and/or ad hoc. Given all that it is IMHO unreasonable to expect the OP to show much more effort. – Jyrki Lahtonen Dec 27 '15 at 22:24
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    @Issac I didn't mean to offend you... A user names Normal Human has actually set up a bot to say the same as I did, and he has way more rep than I, because what I recommend comes directly from the. SE question guide. It is true that many questions show little effort by the person asking the question, but that does not mean that this is what should be happening... Please consult the guide for asking questions on Math.SE to check me on this. I only recommend context because it allows you to find answers faster... It is hard to know how to help someone if they don't tell you much! – Brevan Ellefsen Dec 27 '15 at 22:26
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    I think the question about the post by @BrevanEllefsen is quite legitimate.The question is clear, but the techniques involved in answering such questions can vary from elementary to the frontiers of research. Clarity about the level at which you are working may greatly help those who would like to give constructive answers or comments. Your own work or the context in which the question has arisen will help people to give more constructive answers or to identify the particular points or issues you may have missed. – Mark Bennet Dec 27 '15 at 22:29
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    Your profile shows that you are asking lots of questions about Diophantine equations with degree greater than 2. As @MarcBennet notes in referring to BrevanEllefsen 's comment, these are usually very deep questions. Do you have a particular source for your questions, or are they random? What skills do you have to think about answering them yourself? – Ethan Bolker Dec 27 '15 at 23:40

2 Answers2

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There are several references on integer (and rational) solutions of the equation $$ y^n=1+x+x^2+\cdots +x^{m-1} $$ for $m>2,\; n>1$, e.g. the article of Li Yu and Maohua Le and the references therein; the article of N. Hirata-Kohno and T.N. Shorey of 1996, "On the equation $(x^m-1)/(x-1)=y^q$ and the references therein, in several articles by Y. Bugeaud, and in many other papers of this kind. I think that if a complete answer is possible, then you will find the answer (and a proof) there in the literature . For $(m,n)=(5,5)$ it is your equation. I believe it is your task now, to go through these many cases which have been treated there.

Dietrich Burde
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That $(0,1)$ and $(-1,1)$ are the only two rational points on the curve $y^{5} = x^{4} + x^{3} + x^{2} + x + 1$ was a conjecture made by Lebesgue in 1843, and proven in the 2004 paper of Emmanuel Halberstadt and Alain Kraus (titled "Une conjecture de Lebesgue'' and published in the Journal of the London Math Society, vol. 69, 291-302). Here's a link to the paper from the publisher (although the paper may be behind a paywall for many).

Roughly speaking the technique is to use descent (related to factorization in $\mathbb{Q}(\sqrt{5})$) to relate the rational points on this curve to those on a genus $4$ hyperelliptic curve, then find a map from this curve to an elliptic curve defined over a degree $5$ number field, and apply the method of elliptic curve Chabauty.

Jeremy Rouse
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