For each p ≤ 2 there exist a model M∗ of I∆0(α) which satisfies the Count(p) principle. Furthermore if p contain all prime factors of q there exist n, r ∈ M∗ and a bijective map f ∈ Set(M∗) mapping {1, 2, ..., n} onto {1, 2, ..., n+ qr}. A corollary is a complete classification of the Count(q) versus Count(p) problem. Another corollary solves an open question ([3]). In this note I state and pro...

We present a formalization of the factor theorem for univariate polynomials, also called the (little) Bezout theorem: Let r belong to a commutative ring L and p(x) be a polynomial over L. Then x− r divides p(x) iff p(r) = 0. We also prove some consequences of this theorem like that any non zero polynomial of degree n over an algebraically closed integral domain has n (non necessarily distinct) ...

Abstract Let G be a compact connected Lie group, and let $\operatorname {Hom}({\mathbb {Z}}^m,G)$ the space of pairwise commuting m -tuples in . We study problem which primes $p \operatorname {Z}}^m,G)_1$ , component containing element $(1,\ldots ,1)$ has p -torsion homology. will prove that for $m\ge 2$ homology if only divides order Weyl group $G=SU(n)$ some exceptional groups. also compute t...

The restricted binary partition function bk(n) enumerates the number of ways to represent n as n = 2a0 + 2a1 + · · ·+ 2aj with 0 ≤ a0 ≤ a1 ≤ . . . ≤ aj < k. We study the question of how large a power of 2 divides the difference bk(2n) − bk−2(2n) for fixed k ≥ 3, r ≥ 1, and all n ≥ 1.

Let al <. .. < ak s n be a sequence of integers no one of which divides any other. It 1s not difficult to see that max k-[Et'-) [1]. Assume now that no a i divides the product of two others, then I proved that [2] (r(x) denotes the number of primes c x2/3 (1) n (x) +-l Z < max k (t .9 =) not exceeding x) x 2/3 < e(x) + c 2 (tog =) 2 The proof of both the upper and the lower bound used combinato...

It is well known that, whenever k divides n, the complete k-uniform hypergraph on n vertices can be partitioned into disjoint perfect matchings. Equivalently, set of k-subsets an n-set parallel classes so that each class a partition n-set. This result as Baranyai's theorem, which guarantees existence Baranyai partitions. Unfortunately, proof theorem uses network flow arguments, making this none...

A primitive 3-smooth partition of n is a representation of n as the sum of numbers of the form 2a3b, where no summand divides another. Partial results are obtained in the problem of determining the maximal and average order of the number of such representations. Results are also obtained regarding the size of the terms in such a representation, resolving questions of Erdős and Selfridge. 0. Int...

Note that, by (4 .4) and (4 .5), (7 .6) holds for all nonnegatíve p. Substituting from (7 .6) in (6 .1) and (6 .2) and evaluating coefficients of xm, we obtain the following two identities. r-1 m (p + q)T rP~+4) = pTrpm + ~,(q) + T (P)-L q r, m pq E a l i r-a-1, m-j s-0 j-0 r-1 m-1 pq ~-~ T(P)Tr(9a-l. m-i-1 a L-0 i-0 r-1 m (r + i)Tr m+q) _ (r + 1)Tr + p (r-s)T (P)T (q) a. j r-8-1, M-j 8-0 j-0 r...

Let [Formula: see text] be a Lucas sequence and, for every prime number text], let the rank of appearance in that is, smallest positive integer such divides whenever it exists. Furthermore, an odd integer. Under some mild hypotheses, we prove asymptotic formula primes as text].

Let $X$ be a nonempty set. Denote by $\mathcal{F}^n_k$ the class of associative operations $F\colon X^n\to X$ satisfying condition $F(x_1,\ldots,x_n)\in\{x_1,\ldots,x_n\}$ whenever at least $k$ elements $x_1,\ldots,x_n$ are equal to each other. The $\mathcal{F}^n_1$ said quasitrivial and those $\mathcal{F}^n_n$ idempotent. We show that $\mathcal{F}^n_1=\cdots =\mathcal{F}^n_{n-2}\subseteq\mathc...