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: - 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][1] by D'Aquino.) $I\Delta_0$ seems not to be able to prove that: - 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][2] 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][3] to Parikh; the Pell equation result is [due][4] 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][5] due to Friedman.) [1]: https://www.jstor.org/stable/2275173 [2]: https://www.cs.umd.edu/users/gasarch/TOPICS/provephp.pdf [3]: https://www.math.ucsd.edu/~sbuss/ResearchWeb/parikh/paper.pdf [4]: https://www.sciencedirect.com/science/article/pii/0168007295000186?via%3Dihub [5]: https://en.wikipedia.org/wiki/Elementary_function_arithmetic#Friedman's_grand_conjecture