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In the 1986 book An Introduction to the Theory of Surreal Numbers, Gonshor, on page 117, notes that it is an open problem whether $\omega(\sqrt{2}+1)+1$ is a prime, using the standard definition of integral number he introduces earlier in the chapter.

I'll give a little background, as requested, using Gonshor as a reference.

The surreal numbers are a superclass of the infinitesimals, though I'm not familiar with their exact sizes.

A surreal number is a function from an initial segment of the ordinals into $\{+,-\}$. (An ordinal sequence which terminates)

Then, given two surreals, $a$, $b$, let $a < b$ if $a(\alpha) < b(\alpha)$ where $\alpha$ is the first place they differ, with the convention $- < 0 < +$, where $0$ is an abuse of notation that simply means that the function is undefined at that point. So, $(++) > (+) > (+-)$. Some investigation gives that $0\sim (),\ 1 \sim (+),\ 2 \sim (++), \ldots$ Most of the book is spent on embedding the reals and infinitesimals in the surreals.

Then you can define the generalized integers in the surreals: $a$ is an integer if the exponents in normal form of $a$ are non-negative, and if a $0$ exponent occurs, then the real coefficient is an integer. Normal form is a bit complicated to define, but basically you write numbers in terms of $\omega$ and $\epsilon$ where $\omega$ is the first infinite ordinal, so $(+++...)$ times and $\epsilon = 1/\omega$, which also exists in the surreals. So, $1/3\omega^2+3$ is a generalized integer, and so is $\sqrt{\omega}+2$, but not $0.5+\omega^{-1}$, as both terms violate the definition.

For the definition of prime, $1=(+)$ is still a unit, so primality amounts to proving that a surreal has more than two factors (itself and $1$). $\omega$ is factorizable since $n \in \mathbb{N}$ and $\omega/n$ are both generalized integers.

That trick doesn't work with $\omega(\sqrt{2}+1)+1$.

Is this still an open problem? The book is a couple decades old (picked it up at the library), and searching for surreal numbers doesn't turn up many results.

Thanks

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    $\begingroup$ Ignorant lurkers like me might be able to get more out of reading this if you could sketch more of the background. It's up to you whether you want to spend any time constructing such a sketch, but sometimes doing such a thing as an exercise can clarify one's own thoughts. $\endgroup$ – Ben Crowell Jan 25 at 1:57
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    $\begingroup$ Explanation added $\endgroup$ – Ivan Pong Jan 25 at 14:58
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The answer is no. The factorization is $(\sqrt{\sqrt{2}+1}\omega^{1/2} - \sqrt{2\sqrt{\sqrt{2}+1}}\omega^{1/4}+1)(\sqrt{\sqrt{2}+1}\omega^{1/2} + \sqrt{2\sqrt{\sqrt{2}+1}}\omega^{1/4}+1)$

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    $\begingroup$ This answer reminds me of Frank Nelson Cole (mathoverflow.net/questions/207321/…). How did you find this factorization? $\endgroup$ – Will Brian Mar 29 at 21:16
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    $\begingroup$ I used the initial guess $a+1 \sim (\sqrt{a}+1)^2$ and the applied difference of squares to cancel out the $2\sqrt{a}$ term. $\endgroup$ – Ivan Pong Mar 29 at 21:19
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This is not an answer but it is too long for a comment. I think that the question is still open, and that in general there are no known prime elements with finite support besides primes in $\mathbb{Z}$.


There are a few papers, besides Conway and Gonshor's books, that deal with this type of question. A more general problem is the existence of prime or irreducibles in integer parts of real-closed fields, such as $\mathbf{Oz}$ (the "omnific integers"), which can be written as a Hahn series ring $K((G^{\leq 0}))$ where $G$ is an additively denoted ordered group.

For those with more emphasis on $\mathbf{Oz}$, you can look at Alessandro Berarducci's Factorization in generalized power series (Trans. Amer. Math. Soc. 352 (2000) 553-577, doi:10.1090/S0002-9947-99-02172-8), which I think first found out that one could apply valuation theoretic insight to this question. A more recent paper which in particular gives more recent references for this type of question is Sonia L'Innocente and Vincenzo Mantova's preprint Factorisation theorems for generalised power series, arXiv:1710.07304.

Berarducci gave a positive answer to Conway's conjecture that $\omega+\omega^{\frac{1}{2}}+\omega^{\frac{1}{3}}+ \cdot \cdot \cdot+1$ is irreducible. Daniel Pitteloud subsequently proved that this number is prime in $\mathbf{Oz}$.

L'Innocente and Mantova generalised some results and proved that certain numbers with infinite support as series are prime.

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    $\begingroup$ See my answer. I believe that I just misunderstood Gonshor's statement, as the factorization is fairly trivial. $\endgroup$ – Ivan Pong Mar 29 at 21:27

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