In a recent exchange, kHarsh3715 and I were discussing the potential topics in Number Theory to learn. kHarsh3715, with conviction, deemed **Mobius Inversion**, **Convolutions**, and **Diophantine Equations** as "very useful topics." He further went on to include **Generating Functions**, **Linear Recurrences**, and ventured into the arcane with the **Lucas Theorem**, **Pollard Rho**, and **Primality Testing**.

Baffled by his bold assertions, I simply replied, *"Umnik would disagree"* (Referring to this blog of course: link). This simple statement sparked a fierce debate, escalating to a wager that put our very financial futures at stake. Only the legendary Um_nik himself could resolve this epic conflict and determine who would emerge victorious.

Lucas Theorem, Pollard Rho, and Primality Testing.tf are theseyour profile picture sums up our reaction

I'm afraid your financial future looks bleak my man.

Of course, one can disagree on the definition of "very useful topic". But I agree that all the listed topics are useful, especially the Lucas Theorem which I used dozens of times which is unheard of for such a seemingly niche thing.

Thanks, you just made me rich.

Um_nik and kHarsh3715 got a secret deal against Final_Track (50/50) XD.