189,154 research outputs found

    s(n) An Arithmetic Function of Some Interest, and Related Arithmetic

    Get PDF
    Every integer n > 0 º N defines an increasing monotonic series of integers: n1, n2, ...nk, such that nk = nk +k(k-1)/2. We define as s(m) the number of such series that an integer m belongs to. We prove that there are infinite number of integers with s=1, all of the form 2^t (they belong only to the series that they generate, not to any series generated by a smaller integer). We designate them as s-prime integers. All integers with a factor other than 2 are not s-prime (s>1), but are s-composite. However, the arithmetic s function shows great variability, lack of apparent pattern, and it is conjectured that s(n) is unbound. Two integers, n and m, are defined as s-congruent if they have a common s-series. Every arithmetic equation can be seen as an expression without explicit unknowns -- provided every integer variable can be replaced by any s-congruent number. To validate the equation one must find a proper match of such members. This defines a special arithmetic, which appears well disposed towards certain cryptographic applications

    The Representation of Natural Numbers in Quantum Mechanics

    Full text link
    This paper represents one approach to making explicit some of the assumptions and conditions implied in the widespread representation of numbers by composite quantum systems. Any nonempty set and associated operations is a set of natural numbers or a model of arithmetic if the set and operations satisfy the axioms of number theory or arithmetic. This work is limited to k-ary representations of length L and to the axioms for arithmetic modulo k^{L}. A model of the axioms is described based on states in and operators on an abstract L fold tensor product Hilbert space H^{arith}. Unitary maps of this space onto a physical parameter based product space H^{phy} are then described. Each of these maps makes states in H^{phy}, and the induced operators, a model of the axioms. Consequences of the existence of many of these maps are discussed along with the dependence of Grover's and Shor's Algorithms on these maps. The importance of the main physical requirement, that the basic arithmetic operations are efficiently implementable, is discussed. This conditions states that there exist physically realizable Hamiltonians that can implement the basic arithmetic operations and that the space-time and thermodynamic resources required are polynomial in L.Comment: Much rewrite, including response to comments. To Appear in Phys. Rev.

    Consistency of circuit lower bounds with bounded theories

    Get PDF
    Proving that there are problems in PNP\mathsf{P}^\mathsf{NP} that require boolean circuits of super-linear size is a major frontier in complexity theory. While such lower bounds are known for larger complexity classes, existing results only show that the corresponding problems are hard on infinitely many input lengths. For instance, proving almost-everywhere circuit lower bounds is open even for problems in MAEXP\mathsf{MAEXP}. Giving the notorious difficulty of proving lower bounds that hold for all large input lengths, we ask the following question: Can we show that a large set of techniques cannot prove that NP\mathsf{NP} is easy infinitely often? Motivated by this and related questions about the interaction between mathematical proofs and computations, we investigate circuit complexity from the perspective of logic. Among other results, we prove that for any parameter k≥1k \geq 1 it is consistent with theory TT that computational class C⊈i.o.SIZE(nk){\mathcal C} \not \subseteq \textit{i.o.}\mathrm{SIZE}(n^k), where (T,C)(T, \mathcal{C}) is one of the pairs: T=T21T = \mathsf{T}^1_2 and C=PNP{\mathcal C} = \mathsf{P}^\mathsf{NP}, T=S21T = \mathsf{S}^1_2 and C=NP{\mathcal C} = \mathsf{NP}, T=PVT = \mathsf{PV} and C=P{\mathcal C} = \mathsf{P}. In other words, these theories cannot establish infinitely often circuit upper bounds for the corresponding problems. This is of interest because the weaker theory PV\mathsf{PV} already formalizes sophisticated arguments, such as a proof of the PCP Theorem. These consistency statements are unconditional and improve on earlier theorems of [KO17] and [BM18] on the consistency of lower bounds with PV\mathsf{PV}
    • …
    corecore