共查询到20条相似文献,搜索用时 31 毫秒
1.
M Jungerman 《Journal of Combinatorial Theory, Series B》1975,19(1):69-71
A simpler method for proving when n ≡ 1 (mod 12). 相似文献
2.
C.R.J Clapham 《Journal of Combinatorial Theory, Series B》1973,15(1):74-76
It is shown that the number of triangles in a self-complementary graph with N vertices is at least if N ≡ 0 (mod 4) and at least if N ≡ 1 (mod 4), and that this minimum number can be achieved. 相似文献
3.
Earl S. Kramer 《Discrete Mathematics》1974,8(2):173-180
A t-design (λ, t, d, n) is a system of sets of size d from an n-set S, such that each t subset of S is contained in exactly λ elements of . A t-design is indecomposable (written IND(λ, t, d, n)) if there does not exist a subset ? such that is a (λ, t, d, n) for some λ, 1 ? λ < λ. A triple system is a (λ; 2, 3, n). Recursive and constructive methods (several due to Hanani) are employed to show that: (1) an IND(2; 2, 3, n) exists for n ≡ 0, 1 (mod 3), n ? 4 and n ≡ 7 (designs of Bhattacharya are used here), (2) an IND(3; 2, 3, n) exists for n odd, n ? 5, (3) if an IND(λ, 2, 3, n) exists, n odd, then there exists an infinite number of indecomposable triple systems with that λ. 相似文献
4.
The set S consisting of those positive integers n which are uniquely expressible in the form n = a2 + b2 + c2, , is considered. Since n ∈ S if and only if 4n ∈ S, we may restrict attention to those n not divisible by 4. Classical formulas and the theorem that there are only finitely many imaginary quadratic fields with given class number imply that there are only finitely many n ∈ S with n = 0 (mod 4). More specifically, from the existing knowledge of all the imaginary quadratic fields with odd discriminant and class number 1 or 2 it is readily deduced that there are precisely twelve positive integers n such that n ∈ S and n ≡ 3 (mod 8). To determine those n ∈ S such that n ≡ 1, 2, 5, 6 (mod 8) requires the determination of the imaginary quadratic fields with even discriminant and class number 1, 2, or 4. While the latter information is known empirically, it has not been proved that the known list of 33 such fields is complete. If it is complete, then our arguments show that there are exactly 21 positive integers n such that n ∈ S and n ≡ 1, 2, 5, 6 (mod 8). 相似文献
5.
Criteria are obtained for the quartic residue character of the fundamental unit of the real quadratic field , where q is prime and either q ≡ 7(mod 8), or q ≡ 1(mod 8) and X2 ? 2qY2 = ?2 is solvable in integers X and Y. 相似文献
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7.
Arnošt J.J Heidrich 《Journal of Number Theory》1977,9(4):413-419
Let p be an odd prime and n an integer relatively prime to p. In this work three criteria which give the value of the Legendre symbol () are developed. The first uses two adjacent rows of Pascal's triangle which depend only on p to express () explicitly in terms of the numerically least residues (mod p) of the numbers n, 2n, …, [ or of the numbers . The second, analogous to a theorem of Zolotareff and valid only if p ≡ 1 (mod 4), expresses () in terms of the parity of the permutation of the set {} defined by the absolute values of the numerically least residues of . The third is a result dual to Gauss' lemma which can be derived directly without Euler's criterion. The applications of the dual include a proof of Gauss' lemma free of Euler's criterion and a proof of the Quadratic Reciprocity Law. 相似文献
8.
Charles H.C Little 《Journal of Combinatorial Theory, Series B》1980,29(2):185-194
If S is a collection of circuits in a graph G, the circuits in S are said to be consistently orientable if G can be oriented so that they are all directed circuits. If S is a set of three or more consistently orientable circuits such that no edge of G belongs to more than two circuits of S, then S is called a ring if there exists a cyclic ordering C0, C1,…, Cn ? 1, C0 of the n circuits in S such that if and only if j = i or j ≡ i ? 1 (mod n) or j ≡ i + 1 (mod n). We characterise planar cubic graphs in terms of the non-existence of a ring with certain specified properties. 相似文献
9.
R.J. Cook 《Journal of Number Theory》1979,11(4):505-515
Let where p is a prime ≡ 3 mod 4 and k is an integer ≥ 3. Then S(k) frequently takes large values of each sign. 相似文献
10.
Timothy W Tillson 《Journal of Combinatorial Theory, Series B》1980,29(1):68-74
It is shown that , 2m ≥ 8, can be decomposed into Hamiltonian circuits. A direct construction utilizing difference methods is given for 2m ≡ 0 (mod 4). The case 2m ≡ 2 (mod 4) is handled inductively by means of a construction which shows that admits such a decomposition if does. 相似文献
11.
A triple system is a balanced incomplete block design D(v, k, λ, b, r) with k = 3. Although it has been shown that triple systems exist for all values of the parameters satisfying the necessary conditions: direct methods (nonrecursive) of construction are not available in general. In this paper we give a direct method to construct a triple system for all values of the parameters satisfying the necessary conditions. 相似文献
12.
Marvin I. Knopp 《Journal of Number Theory》1980,12(1):2-9
If h, k ∈ Z, k > 0, the Dedekind sum is given by , with , . The Hecke operators Tn for the full modular group SL(2, Z) are applied to log η(τ) to derive the identities (n ∈ Z+) , where (h, k) = 1, k > 0 and σ(n) is the sum of the positive divisors of n. Petersson had earlier proved (1) under the additional assumption k ≡ 0, h ≡ 1 (mod n). Dedekind himself proved (1) when n is prime. 相似文献
13.
Let F be a field and A a maximal commutative subalgebra of the full matrix algebra Mn(F). It is shown that dim A > (2n) ? 1. It is also shown that if the radical of A has cube zero, then dim A ? [3n ? 4], and that this result is best possible for infinitely many natural numbers n. 相似文献
14.
Paul Terwilliger 《Discrete Mathematics》1982,41(3):295-302
We find lower bounds on eigenvalue multiplicities for highly symmetric graphs. In particular we prove:Theorem 1. If Γ is distance-regular with valency k and girth g (g?4), and λ (λ≠±?k) is an eigenvalue of Γ, then the multiplicity of λ is at least if g≡0 or 1 (mod 4), if g≡2 or 3 (mod 4) where [ ] denotes integer part. Theorem 2. If the automorphism group of a regular graph Γ with girth g (g?4) and valency k acts transitively on s-arcs for some s, , then the multiplicity of any eigenvalue λ (λ≠±?k) is at least if s is even, if s is odd. 相似文献
15.
Francisco Thaine 《Journal of Number Theory》1985,20(2):128-142
Let p be an odd prime and suppose that for some a, b, c ? \p we have that ap + bp + cp = 0. In Part I a simple new expression and a simple proof of the congruences of Mirimanoff which appeared in his papers of 1910 and 1911 are given. As is known, these congruences have Wieferich and Mirimanoff criteria (2p ? 1 ≡ 1 mod p2 and 3p ? 1 ≡ 1 mod p2) as immediate consequences. Mirimanoff's congruences are expressed in the form of polynomial congruences , 1 ≤ m ≤ p ? 1, and these polynomials Pm(X) are characterized by means of simple relations. In Part II a complement to Kummer-Mirimanoff congruences is given under the hypothesis that p does not divide the second factor of the class number of the p-cyclotomic field. 相似文献
16.
This paper deals with two topics, namely, frames and pairwise balanced designs (PBD's). Frames, which were introduced by W.D. Wallis for the construction of (skew) Room squares, are shown to exist for most orders congruent to 1 (mod 4). This result relies heavily on the existence of PBD's since the set F = {v | there is a frame of order v] is shown to be PBD-closed. By employing a generalization of the usual recursive construction for PBD's, it is shown that {5, 9, 13, 17}?{5, 9, 13}∪{69, 77, 97, 137, 237, 277, 317, 377, 569}?{n | n 1 (mod 4), n>0}?{29, 33, 49, 57, 93, 129, 133}, where (K) denotes the set of orders of PBD's of index one having block-sizes from the set K. Frames of orders 5, 9, 13 and 17 are exhibited which immediately implies that F?{5, 9, 13, 17}. D.R. Stinson and W.D. Wallis have shown that {29, 49}?F. Thus there is a frame of order υ for every positive integer υ congruent to 1 (mod 4) with the possible exceptions of υ ? {33, 57, 93, 133}. 相似文献
17.
Ching-Shui Cheng 《Journal of Combinatorial Theory, Series A》1981,31(2):219-221
It is shown that if s is a prime or a prime power with s ≡ 3 (mod 4), then there is an array of s2 symbols whose rows and columns together form a balanced incomplete block design. 相似文献
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In this paper, we prove that admits a resolvable decomposition into TT3 or C3 if and only if n ≡ 0 (mod. 3), n ≠ 6. 相似文献