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We establish a new bridge between propositional logic and elementary numbertheory. The main objects are "minimally unsatisfiable clause-sets", short"MUs", unsatisfiable conjunctive normal forms rendered satisfiable byelimination of any clause. In other words, irredundant coverings of the booleanhypercube by subcubes. The main parameter for MUs is the "deficiency" k, thedifference between the number of clauses and the number of variables (thedifference between the number of elements in the covering and the dimension ofthe hypercube), and the fundamental fact is that k >= 1 holds.  A "full clause" in an MU contains all variables (corresponding to a singletonin the covering). We show the lower bound S_2(k) <= FCM(k), where FCM(k) is themaximal number of full clauses in MUs of deficiency k, while S_2(k) is thesmallest n such that 2^k divides n!.  The proof rests on two methods: On the logic-combinatorial side, applyingsubsumption resolution and its inverse, a fundamental method since Boole in1854 introduced the "expansion method". On the arithmetical side, analysingcertain recursions, combining an application-specific recursion with arecursion from the field of meta-Fibonacci sequences (indeed S_2 equals twicethe Conolly sequence).  A further tool is the consideration of unsatisfiable "hitting clause-sets"(UHITs), special cases of MUs, which correspond to the partitions of theboolean hypercube by subcubes; they are also known as orthogonal or disjointDNF tautologies. We actually show the sharper lower bound S_2(k) <= FCH(k),where FCH(k) is the maximal number of full clauses in UHITs of deficiency k. Weconjecture that for all k holds S_2(k) = FCH(k), which would establish asurprising connection between the extremal combinatorics of (un)satisfiabilityand elementary number theory.  We apply the lower bound to analyse the structure of MUs and UHITs.

Parameters for minimal unsatisfiability: Smarandache primitive numbers and full clauses