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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 (++), ...$ 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 $0.5+\omega^{-1}$, as both terms violate the definition.

For the defintion 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 factorizeable 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

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

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.

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

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 (++), ...$ 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 $0.5+\omega^{-1}$, as both terms violate the definition.

For the defintion 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 factorizeable 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

In the 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.

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

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.

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