We introduce a new approach to a century old assumption which enhances not only planetary interior calculations but also high pressure material physics. We show that the polytropic index is the derivative of the bulk modulus with respect to pressure. We then augment the traditional polytrope theory by including a variable polytrope index within the confines of the Lane-Emden differential equation. To investigate the possibilities of this method we create a high quality universal equation of state, transforming the traditional polytrope method to a tool with the potential for excellent predictive power. The theoretical foundation of our equation of state is the same elastic observable which we found equivalent to the polytrope index, the derivative of the bulk modulus with respect to pressure. We calculate the density-pressure of six common materials up to 10 18 Pa, mass-radius relationships for the same materials, and produce plausible density-radius models for the rocky planets of our solar system. We argue that the bulk modulus and its derivatives have been under utilized in previous planet formation methods. We constrain the material surface observables for the inner core, outer core, and mantle of planet Earth in a systematic way including pressure, bulk modulus, and the polytrope index in the analysis. We believe this variable polytrope method has the necessary apparatus to be extended further to gas giants and stars. As supplemental material we provide computer code to calculate multi-layered planets.

In recent years, the congruence [Formula: see text] first discovered by the last author has been generalized by either increasing the number of indices and considering the corresponding super congruences, or by considering the alternating version of multiple harmonic sums. In this paper, we prove a family of similar super congruences modulo prime powers [Formula: see text] with the indices summing up to [Formula: see text] where [Formula: see text] is coprime to [Formula: see text], and where all the indices are also coprime to [Formula: see text].

P rogramming has rapidly become an essential tool for nearly all students in the physical sciences. One of the most widely used languages in this realm is Python, which has swiftly gained popularity due to its readability and intuitive syntax. The core philosophy or "Zen of Python" dictates, "simple is better than complex, complex is better than complicated" (https://www.python. org/dev/peps/pep-0020/). Because of this motto, learning how to use Python effectively on your own is a very doable task, given the right resources. This is where A Student's Guide to Python for Physical Modeling by Jesse M. Kinder and Philip Nelson comes in. The text serves as an excellent stepping stone into the world of using Python in computational science for undergraduate students