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51.
Based on results of Weil and of Burgess, we have obtained a boundK(l) such that all primesp K(l) have a sequence of at leastl consecutive quadratic residues and a sequence of at leastl consecutive nonresidues in the interval [1,p – 1]. The bound forl=9 being 414463, we have computed, for primes less than 420000, the lengths of the longest sequences of consecutive residues and of nonresidues. We present these data and make some observations concerning them. One of the observations is that there is an observed difference in the length of the maximal sequence between primes congruent to 1 (mod 4) and primes congruent to 3 (mod 4). 相似文献
52.
Let q and p be prime with q = a2 + b2 ≡ 1 (mod 4), a ≡ 1 (mod 4), and p = qf + 1. In the nineteenth century Cauchy (Mém. Inst. France17 (1840), 249–768) and Jacobi (J. für Math.30 (1846), 166–182) generalized the work of earlier authors, who had determined certain binomial coefficients (mod p) (see H. J. S. Smith, “Report on the Theory of Numbers,” Chelsea, 1964), by determining two products of factorials given by Πkkf! (mod p = qf + 1) where k runs through the quadratic residues and the quadratic non-residues (mod q), respectively. These determinations are given in terms of parameters in representations of ph or of 4ph by binary quadratic forms. A remarkable feature of these results is the fact that the exponent h coincides with the class number of the related quadratic field. In this paper C. R. Mathews' (Invent. Math.54 (1979), 23–52) recent explicit evaluation of the quartic Gauss sum is used to determine four products of factorials (mod p = qf + 1, q ≡ 5 (mod 8) > 5), given by Πkkf! where k runs through the quartic residues (mod q) and the three cosets which may be formed with respect to this subgroup. These determinations appear to be considerably more difficult. They are given in terms of parameters in representations of 16ph by quaternary quadratic forms. Stickelberger's theorem is required to determine the exponent h which is shown to be closely related to the class number of the imaginary quartic field Q(i√2q + 2a√q), q = a2 + b2 ≡ 5 (mod 8), a odd. 相似文献
53.
Vandana Tripathi Lagy T Baby PV Madhusudhana Rao SK Hui R Singh JJ Das P Sugathan N Madhavan AK Sinha 《Pramana》1999,53(3):535-539
The ground state and excited state transfer yields for the 2-neutron pickup channel in the 28Si+68Zn system have been measured explicitly. The recoil mass separator at the nuclear Science Centre, New Delhi was used for the
measurement. A NaI(T1) detector was used for detecting the deexcitation γ’s from the transfer products. The kinematic coincidence technique was employed for the transfer measurement. Simplified coupled
channels calculations show that out of all transfer channels the major contribution to the sub-barrier enhancement comes from
the ground state 2 neutron pickup channel with a ground state Q-value of+1.83 MeV. 相似文献
54.
A comparative study of the thermomagnetic memory effects of J
c in two weakly pinned low T
c superconductors, Ca3Rh4Sn13 (CaRhSn) and Yb3Rh4Sn13 (YbRhSn), is presented. In both the systems, the peak effect (PE) phenomenon appears as an order-disorder transformation
through stepwise amorphization of the flux line lattice (FLL). However, in CaRhSn. we can witness another disorder-driven
transition (Bragg glass (BG) to a vortex glass (VG) in a distinct manner as in a single crystal of high T
c YBa2Cu3O7-δ for H‖c. 相似文献
55.