arXiv subjects

Bernd R. Schuh

Publications and source records attributed to Bernd R. Schuh.

11 recordsLinked to original sources

arXiv2025

The Erdös-Straus Conjecture and Pythagorean Primes

The Diophantine equation 4/n=1/x+1/y+1/z for a Pythagorean prime n is split into two independent Diophantine equations, which correspond to two different types of solution. The solvability of these equations forces certain restrictions on allowed Pythagorean primes. Empirical evidence suggests that these restrictions hold for all Pythagorean primes, which I state as two independent conjectures. One can be formulated as follows: every Pythagorean prime can be written as p=(4ab-1)(4c-1)-4abb/d, where a, b, c are natural numbers and d is a divisor of ab. The second conjecture reads: every Pythagorean prime can be written as p=(4ab-1)(4c-1)-4ac, where a, b, c are natural numbers. I give a new straightforward plausibility for the latter conjecture (which has been formulated independently by other authors) and I outline a practicle and effective algorithm to determine a,b,c for a given p.

math.GM↗

arXiv2024

Unique polynomial solution of $m/n=1/x+1/y+1/z$ for $n \equiv b {\rm mod}\, a$ if $(a,m)=1$

Necessary and sufficient conditions for the existence of an integer solution of the diophantine equation $m/n=1/x(λ)+1/y(λ)+1/z(λ)$ with $n=b+aλ$ are explicitly given for a,b coprime and a not a multiple of m . The solution has the form $x(λ)=kn(λ)$, $y(λ)=n(λ)(s+rλ)$, $z(λ)=(kl/r)(s+rλ)$ where parameters $k,l,s,r\in \mathbb{Z}$ obey certain conditions depending on $a,b$. The conditions imply restrictions for some choices of $a,b$ which differ from the ones known in the case $m=4$. E.g., the modulus must be of the form $l(mk-1)$. One can also deduce that primes of the form $1+4K$ are excluded as modulus. Also if $a=p\ne m$ is prime and $b=a+1$, i.e., $n\equiv 1{\rm mod}\, p$, polynomial solutions are shown to be impossible. All results are valid for integers $m \ge 4$.

math.GM↗

arXiv2018

Sub-exponential complexity of regular linear CNF formulas

The study of regular linear conjunctive normal form (LCNF) formulas is of interest because exact satisfiability (XSAT) is known to be NP-complete for this class of formulas. In a recent paper it was shown that the subclass of regular exact LCNF formulas (XLCNF) is of sub-exponential complexity, i.e. XSAT can be determined in sub-exponential time. Here I show that this class is just a subset of a larger class of LCNF formulas which display this very kind of complexity. To this end I introduce the property of disjointedness of LCNF formulas, measured, for a single clause C, by the number of clauses which have no variable in common with C. If for a given LCNF formula F all clauses have the same disjointedness d we call F d-disjointed and denote the class of such formulas by dLCNF. XLCNF formulas correspond to the special cased=0. One main result of the paper is that the class of all monotone l-regular LCNF formulas which are d-disjointed, with d smaller than some upper bound D, is of sub-exponential complexity. This result can be generalized to show that all monotone, l-regular LCNF formulas F which have a bounded mean disjointedness, are of sub-exponential XSAT-complexity, as well.

cs.CC↗

arXiv2017

A criterion for "easiness" of certain SAT problems

A generalized 1-in-3SAT problem is defined and found to be in complexity class P when restricted to a certain subset of CNF expressions. In particular, 1-in-kSAT with no restrictions on the number of literals per clause can be decided in polynomial time when restricted to exact READ-3 formulas with equal number of clauses (m) and variables (n), and no pure literals. Also individual instances can be checked for easiness with respect to a given SAT problem. By identifying whole classes of formulas as being solvable efficiently the approach might be of interest also in the complementary search for hard instances.

cs.CC↗

arXiv2017

Polynomial time estimates for #SAT

Limits on the number of satisfying assignments for CNS instances with n variables and m clauses are derived from various inequalities. Some bounds can be calculated in polynomial time, sharper bounds demand information about the distribution of the number of unsatisfied clauses. Quite generally, the number of satisfying assignments involve variance and mean of this distribution. For large formulae, m>>1, bounds vary with 2**n/n, so they may be of use only for instances with a large number of satisfying assignments.

cs.CC↗

arXiv2014

Easy/Hard Transition in k-SAT

A heuristic model procedure for determining satisfiability of CNF-formulae is set up and described by nonlinear recursion relations for m (number of clauses), n (number of variables) and clause filling k. The system mimicked by the recursion undergoes a sharp transition from bounded running times (easy) to uncontrolled runaway behaviour (hard). Thus the parameter space turns out to be separated into regions with qualitatively different efficiency of the model procedure. The transition results from a competition of exponential blow up by branching versus growing number of orthogonal clauses.

cs.CC↗

arXiv2014

SAT for pedestrians

The aim of this short note is mainly pedagogical. It summarizes some knowledge about Boolean satisfiability (SAT) and the P=NP? problem in an elementary mathematical language. A convenient scheme to visualize and manipulate CNF formulae is introduced. Also some results like the formulae for the number of unsatisfied clauses and the number of solutions might be unknown.

cs.CC↗

arXiv2012

Phase Transition in Unrestricted Random SAT

For random CNF formulae with m clauses, n variables and an unrestricted number of literals per clause the transition from high to low satisfiability can be determined exactly for large n. The critical density m/n turns out to be strongly n-dependent, ccr = ln(2)/(1-p)^^n, where pn is the mean number of positive literals per clause.This is in contrast to restricted random SAT problems (random K-SAT), where the critical ratio m/n is a constant. All transition lines are calculated by the second moment method applied to the number of solutions N of a formula. In contrast to random K-SAT, the method does not fail for the unrestricted model, because long range interactions between solutions are not cut off by disorder.

cs.CC↗

arXiv2010

A Real World Mechanism for Testing Satisfiability in Polynomial Time

Whether the satisfiability of any formula F of propositional calculus can be determined in polynomial time is an open question. I propose a simple procedure based on some real world mechanisms to tackle this problem. The main result is the blueprint for a machine which is able to test any formula in conjunctive normal form (CNF) for satisfiability in linear time. The device uses light and some electrochemical properties to function. It adapts itself to the scope of the problem without growing exponentially in mass with the size of the formula. It requires infinite precision in its components instead.

cs.LO↗

arXiv2009

Logical Primes, Metavariables and Satisfiability

For formulas F of propositional calculus I introduce a "metavariable" MF and show how it can be used to define an algorithm for testing satisfiability. MF is a formula which is true/false under all possible truth assignments iff F is satisfiable/unsatisfiable. In this sense MF is a metavariable with the "meaning" 'F is SAT'. For constructing MF a group of transformations of the basic variables ai is used which corresponds to 'flipping" literals to their negation. The whole procedure corresponds to branching algorithms where a formula is split with respect to the truth values of its variables, one by one. Each branching step corresponds to an approximation to the metatheorem which doubles the chance to find a satisfying truth assignment but also doubles the length of the formulas to be tested, in principle. Simplifications arise by additional length reductions. I also discuss the notion of "logical primes" and show that each formula can be written as a uniquely defined product of such prime factors. Satisfying truth assignments can be found by determining the "missing" primes in the factorization of a formula.

math.LO↗

arXiv2009

Algebraic Properties of Propositional Calculus

In this short note we relate some known properties of propositional calculus to purely algebraic considerations of a Boolean algebra. Classes of formulas of propositional calculus are considered as elements of a Boolean algebra. As such they can be represented by uniquely defined elements of this algebra which we call "logical primes". The algebraic notations appear useful because they make it possible to derive well known properties of propositional calculus by simple calculations or to substitute lengthy logical considerations by schematic algebraic manipulations.

math.LO↗