arXiv subjects

N. F. Benschop

Publications and source records attributed to N. F. Benschop.

8 recordsLinked to original sources

arXiv2009

Additive structure of Z(.) mod m_k (squarefree) and Goldbach's Conjecture

The product m_k of the first k primes (2..p_k) has neighbours m_k +/- 1 with all prime divisors beyond p_k, implying there are infinitely many primes [Euclid]. All primes between p_k and m_k are in the group G_1 of units in semigroup Z_{m_k}(.) of mutiplication mod m_k. Due to the squarefree modulus Z_{m_k} is a disjoint union of 2^k groups, with as many idempotents - one per divisor of m_k, which form a Boolean lattice BL. The generators of Z_{m_k} and the additive properties of its lattice are studied. It is shown that each complementary pair in BL adds to 1 mod m_k and each even idempotent e in BL has successor e+1 in G_1. It follows that G_1+G_1 \equiv E, the set of even residues in Z_{m_k}, so each even residue is the sum of two roots of unity, proving "Goldbach for Residues" mod m_k ("GR"). . . . Induction on k by extending residues mod m_k with "carry" a < p_{k+1} of weight m_k, yields a prime sieve for integers. Failure of Goldbach's Conjecture ("GC") for some 2n contradicts GR(k) for some k. By Bertrand's Postulate (on prime i 1) successive 2n are in overlapping intervals, while the smallest composite unit in G_1 mod m_k is p_{k+1}^2, yielding "GC": Each 2n > 4 is the sum of two odd primes.

math.GM↗

arXiv2001

On primitive roots of unity, divisors of p+/-1, Wieferich primes, and quadratic analysis mod p^3

Primitive roots of 1 mod p^k (k>2 and odd prime p) are sought, in cyclic units group G_k = A_k B_k mod p^k, coprime to p, of order (p-1)p^{k-1}. 'Core' subgroup A_k has order p-1 independent of k, and p+1 generates 'extension' subgroup B_k of all p^{k-1} residues 1 mod p. Divisors r,t of powerful generator p-1=rs=tu of \pm B_k mod p^k, and of p+1, are investigated as primitive root candidates. Fermat's Small Theorem: x^{p-1} \e 1 mod p for 0 2}. And for prime p: 2^p !=2 and 3^p != 3 (mod p^3). Re: Wieferich primes [4] and FLT case_1. Conj: at least one divisor of p \pm 1 is a semi primitive root of 1 mod p^k. -- (paper withdrawn, re thm2.2)

math.GM↗

arXiv2001

Triplets and Symmetries of Arithmetic mod p^k

The finite ring Z_k = Z(+,.) mod p^k of residue arithmetic with odd prime power modulus is analysed. The cyclic group of units G_k in Z_k(.) has order (p-1)p^{k-1}, implying product structure G_k = A_k B_k. Here core A_k of order p-1 is an extension for k >1 of Fermat's Small Theorem (FST*), where n^p == n (mod p^k) for each core residue, while extension subgroup B_k has order p^{k-1}. It is shown that each subgroup S >1 of core A_k has zero sum, and that p+1 generates subgroup B_k of all n == 1 (mod p) in G_k. The p-th power residues n^p mod p^k in G_k form an order |G_k|/p subgroup F_k, with |F_k|/|A_k| = p^{k-2}, so F_k properly contains core A_k for k >2. By quadratic analysis (mod p^3) rather than linear analysis (mod p^2, re Hensel's lemma [5]), the additive structure of subgroups G_k and F_k is derived. ... Successor function S(n)=n+1 combines with the two arithmetic symmetries -n (complement) and 1/n (inverse) to yield the "triplet structure" of G_k : three inverse pairs {n_i, 1/(n_i)} with (n_i)+1 = - 1/n_{i+1} (mod p^k), with indices mod 3, and product n_0.n_1.n_2 = 1 mod p^k. In case n_0 = n_1 = n_2 = n this reduces to the cubic root solution n+1 = -(1/n) = -(n^2) (mod p^k, p=1 mod 6). The property "EDS" of exponent p distributing over a sum of core residues: (x+y)^p == x+y == x^p + y^p (mod p^k), is employed to derive the known FLT inequality for integers. In other words, to any FLT(mod p^k) equivalence for k digits correspond p-th power integers of pk digits, and the (p-1)k "carries" make the difference, representing the sum of mixed-terms in the binomial expansion.

math.GM↗

arXiv2001

A new Binary Number Code and a Multiplier, based on 3 as semi-primitive root of 1 mod 2^k

The powers of 3 generate half of the odd residues mod 2^k (k>2), and a sign change yields the other half. In other words: 3 is a semi-primitive root of 1 mod 2^k (k>2). Hence each k-bit residue is n = +/- 3^i.2^j mod 2^k, with unique non-neg exponent pair: i<2^{k-2} and j<k. -- A new "dual base logarithmic" binary number code (bases 2 and 3) employs this property. This (binary) log-code [s,i,j] - where s is the corresponding sign, simplifies binary multiplication by translating it to addition of the exponents of 2 and 3, and XOR of the signs involved. -- Patent US-5923888 (13jul99)

math.GM↗

arXiv2001

Symmetric Logic Synthesis with Phase Assignment

Decomposition of any Boolean Function BF_n of n binary inputs into an optimal inverter coupled network of Symmetric Boolean functions SF_k (k \leq n) is described. Each SF component is implemented by Threshold Logic Cells, forming a complete and compact T-Cell Library. Optimal phase assignment of input polarities maximizes local symmetries. The "rank spectrum" is a new BF_n description independent of input ordering, obtained by mapping its minterms onto an othogonal n \times n grid of (transistor-) switched conductive paths, minimizing crossings in the silicon plane. Using this ortho-grid structure for the layout of SF_k cells, without mapping to T-cells, yields better area efficiency, exploiting the maximal logic path sharing in SF's. Results obtained with an optimization tool "Ortolog" based on these concepts, for very fast O(n^2) detecting and enhancing local symmetries of a BF_n, are reported. Relaxing symmetric- to planar- Boolean functions is sketched, to improve low- symmetry BF decomposition.

math.GM↗

arXiv2001

Finite Semigroups of Constant Rank, and the five Basic State Machine types

Constant Rank (CR) state machines play an important role in the general structure theory of Finite State Machines. A machine is of constant rank if each input and input-sequence maps the state set onto the same number of next states. CR-machines are analysed via their sequential closure (semigroup), which is a simple semigroup, thus: a semi- direct product (L \times R)*G of a left- and a right-copy semigroup, and a group. . . . So in general a CR-machine is a composition of: a branch-, a reset- and a permutation machine, which are three of the five basic types of state machines, to be derived.

math.GM↗

arXiv2001

Powersums representing residues mod p^k, from Fermat to Waring

The ring Z_k(+,.) mod p^k with prime power modulus (prime p>2) is analysed. Its cyclic group G_k of units has order (p-1)p^{k-1}, and all p-th power n^p residues form a subgroup F_k with |F_k|=|G_k|/p. The subgroup of order p-1, the core A_k of G_k, extends Fermat's Small Theorem (FST) to mod p^{k>1}, consisting of p-1 residues with n^p = n mod p^k. The concept of "carry", e.g. n' in FST extension n^{p-1} = n'p+1 mod p^2, is crucial in expanding residue arithmetic to integers, and to allow analysis of divisors of 0 mod p^k. . . . . For large enough k \geq K_p (critical precison K_p < p depends on p), all nonzero pairsums of core residues are shown to be distinct, upto commutation. The known FLT case_1 is related to this, and the set F_k + F_k mod p^k of p-th power pairsums is shown to cover half of units group G_k. -- Yielding main result: each residue mod p^k is the sum of at most four p-th power residues. Moreover, some results on the generative power (mod p^{k>2}) of divisors of p^2-1 are derived. -- [Publ.: "Computers and Mathematics with Applications", V39 N7-8 (Apr.2000) p253-261]

math.GM↗

arXiv2001

On Fermat's marginal note: a suggestion

A suggestion is put forward regarding a partial proof of FLT(case1), which is elegant and simple enough to have caused Fermat's enthusiastic remark in the margin of his Bachet edition of Diophantus' "Arithmetica". It is based on an extension of Fermat's Small Theorem (FST) to mod p^k for any k>0, and the cubic roots of 1 mod p^k for primes p=1 mod 6. For this solution in residues the exponent p distributes over a sum, which blocks extension to equality for integers, providing a partial proof of FLT case1 for all p=1 mod 6. This simple solution begs the question why it was not found earlier. Some mathematical, historical and psychological reasons are presented. . . . . In a companion paper, on the triplet structure of Arithmetic mod p^k, this cubic root solution is extended to the general rootform of FLT (mod p^k) (case1), called "triplet". While the cubic root solution (a^3=1 mod p^k) involves one inverse pair: a+a^{-1} = -1 mod p^k, a triplet has three inverse pairs in a 3-loop: a+b^{-1} = b+c^{-1} = c+a^{-1} = -1 (mod p^k) where abc = 1 (mod p^k), which reduces to the cubic root form if a=b=c (\neq 1) mod p^k. The triplet structure is not restricted to p-th power residues (for some p \geq 59) but applies to all residues in the group G_k(.) of units in the semigroup of multiplication mod p^k.

math.HO↗