What can $I\Delta_0$ prove? What combinatorial and number-theoretic propositions can $I\Delta_0$ prove? Obviously there are an infinitude of them, but what are some well known theorems that can be proved in $I\Delta_0$, if any?
 A: Since no one else is biting, I'll answer, and thanks to comments I now this is accurate:
$I\Delta_0$ can prove several basic theorems:

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*Every square equals 0 or 1 mod 4

*No prime has a rational square root

*The only solutions to $x^3+y^3=z^3$ or $x^4+y^4=z^4$ are trivial

*Every $x$ is divisible by a prime $p$ with $p \le x$
(The standard proofs can be reproduced in $I\Delta_0$, since they do not require any lists or sequences or products thereof. The last claim is proved in $I\Delta_0$ in a paper by D'Aquino.)
$I\Delta_0$ seems not to be able to prove that:

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*there are arbitrarily large primes

*every prime of the form $4m+1$ can be written as $a^2+b^2$
(The first is a well-known open problem due to Wilkie)
$I\Delta_0$ cannot prove that:

*

*the functions $x^{\log x}$, $x!$, or $x^y$ are total

*there are solutions to the Pell equation $x^2-Ny^2=1$
(The $x^{\log x}$ is due to Parikh; the Pell equation result is due to D'Aquino.)
But $I\Delta_0(exp)$, i.e. the theory $I\Delta_0$ in the language with exponentiation, seems  to prove

*

*every theorem published in the Annals of Mathematics whose statement involves only finitary mathematical objects

(This is a well-known conjecture due to Friedman.)
