Divisor

About: Divisor is a research topic. Over the lifetime, 2462 publications have been published within this topic receiving 21394 citations. The topic is also known as: factor & submultiple.

CMOS Implementation of a hybrid radix-4 divider

Abstract: A 1.2 ?m CMOS combinational implementation of a new hybrid radix-4 division algorithm is presented. The algorithm is named hybrid because the dividend, the quotient, and the remainder are represented using the signed-digit-set {2,1,0,1,2}; while the divisor is represented using the conventional digit-set {0, 1, 2, 3}. The divider requires the divisor Y to be pre-scaled to the range 1 ? Y ≪ 1 + 118. For 16 bit accuracy, it is about 50 % less expensive but 12 % slower than a corresponding radix-2 divider.

Bi-unitary perfect polynomials overGF(g)

Abstract: This paper continues the author's excursions into the arithmetic of polynomials over finite fields. For monic polynomials A, B e GF[q, x] where p is a prime, q=pdand d ⩾ 1: The divisor B of A is a bi-unitary divisor of A provided 1 is the greatest common unitary divisor of the polynomials B and A/B, and we say that A is bi-unitary perfect (b.u.p.) over GF(q) provided A equals the sum σ**(A) of the distinct bi-unitary divisors of A in GF[q, x]. A diversity of b.u.p. polynomials over GF(q) is found, some of which are neither perfect nor unitary perfect. For p > 2 we can only conjecture a characterisation of the b.u.p. polynomials which split in GF[p, x], so several open questions remain. Examples of non-splitting b.u.p. polynomials over GF(p) are given for p=2, 3, 5 which, in turn, allow the construction of such examples over GF(pd) for these p.

On Ribet's isogeny for $J_0(65)$

Abstract: Let $J^{65}$ be the Jacobian of the Shimura curve attached to the indefinite quaternion algebra over $\mathbb{Q}$ of discriminant $65$. We study the isogenies $J_0(65)\rightarrow J^{65}$ defined over $\mathbb{Q}$, whose existence was proved by Ribet. We prove that there is an isogeny whose kernel is supported on the Eisenstein maximal ideals of the Hecke algebra acting on $J_0(65)$, and moreover the odd part of the kernel is generated by a cuspidal divisor of order $7$, as is predicted by a conjecture of Ogg.

On Class Numbers of Hyperelliptic Function Fields-3-

Abstract: Let F =GF (p) be a finite prime field of characteristic p≠2.Let K=F (x, y) be an algebraic functicon field over F defined by an equation y2 = xn-a (a≠0,a∈F), where n means an odd number so that n>1 and p?n. Let h be the class number of K and g the genus of K. Then, it is obvious that h=p+1 if n=3 and p≡2 mod 3. This particular fact can be generally expressed as follows; Given n, there exists an integer c such that h = (p + 1) g whenever p≡c mod n. In this note, it is shown that this generalization is true in the particular case of n = 5 and of n=7. 1.Jntroduction. Let F=GF(P)be、a finite prime field of characteristic p≠2. Let n be an odd number satisfying n>l and pyn. Throughout this note,K=F (x,y)means an algebraic function field over F defined by an equation y2=Xn一一a (a≠0,a∈F).If we denoteby g the genus of K,itis obvious that g=(n-1)/2. Let h be the class nllmber of K,i・e▲,the order of the finite group of divisor classes of degree zero. We willthen discuss the followlng queStion; Does there exist aninteger cwhich depends only on n such that h=(P+1)g whenever P≡C mOd n? In the case of n=3,We had an answerin the affirmative.([5],Theoreml (i)). When h=Pg+l then the similar q11eStion was discussed.([6]).In this note,We Wish to give an answer onlyin the case of n=5and of n=7.In doing so,We Will recall a method of estimating class numbers of algebraic function fields without proofs but with references. Let L(u)=1+alu+a2u2+...+Pg-2a2u2g 2+Pg lalu2g-1+Pgu2g be the L-function of K.Thenitis obvious that h=L(1). Asis well known,the explicit expression for coefficients al,a2and a3Can be putin the form 2 Tadashi NASHIO a, = Nl ~ ( p + I ) (1) 2a2 = N12 ~ (2 p + I ) N1 + 2N2 + 2 p 6a3 = N,a _ 3 pN12 + (3 p l) N1 ~ 6 ( p + I ) N2 + 6NIN2 + 6N3 where Nd means the number of prime divisors of degree d of K. (M.L.Madan and C.S.Queen [2], p. 427). Thus , for our present purpose , it is enough to compute Nd . Since Nd depends on the number of prime divisors of degree one in some constant field extensions of K, in S 2 we will investigate the number N(Kd) of prime divisors of degree one of an algebraic function field Kd Over a finite field Fd . In S 3 , we will compute h in the case of n=5 and of n=7. 2. The number of prime divisors of degree one. Let Kd be the constant field extension of K of degree d and let Fd be the finite field GF (pd) with pd elements . Let us denote by N(Kd) the number of prime divisors of degree one of Kd . We will then consider N(Kd) under the assumption p~~-1 mod n . THEOREM l. If d~;1 mod 2 and p~~-1 mod n, then the equality N(Kd) =pd+1 holds . PROOF. By the definition of N(Kd) , we have N(Kd) =1+# {(a, p) EFdXFd ; p2=a a} So we need to estimate the last term in this formula . Since our assumtions d~~I mod 2 and p~~-1 mod n lead to (pd-1,n) =1, we can get Fdn=Fd in view of the fact that F*=Fd{ O } is a cyclic group of order pd-1 . This implies that #{(a,O)EFdXFd ; a"=a} =1 and #{(a,p)EFdXFd ; p~0 , p2=cr"-a} =pd-1. Therefore we have N(Kd) =1 + l+pd -1 = pd +1 . We will now consider N(Kd) in the cas9 of d=2. THEOREM 2. If p~E-1 mod n and n=2g+1, then N(K2)=p'+2gp+1 holds. We will prepare some lemmas for the proof of this theorem. The following lemma will be proved on the basis of the properties of Hasse-Witt matrices of the algebraic function field Kd Over Fd . LEMMA I . If p~~ I mod n, then N (Kd) ~= I mod p for an arbitrary positive integer d . PROOF. For O~u,v~;g I , Iet A~,, be the coefficient of x"+* in the following pol ynomial I~((x"-a) (p~')/2x"'1)=1~( ~] (p~1)/2 (p-1)/2 (_a) (p~1) 2 'x"""'t) where I~ means ( ) .=0 r the p~1 Iinear operator satisfying r x"/p if p j w I~ (x") = The matrix A=(A~,.) is called the Hasse-Witt matrix. (L.Miller [3]) . Since it is easy in our case to show that nr+u+1~p(v+1) for every O~u, v~g-1, we On Class Numbers of Hyperelliptic Function Fields,五 3 haveん,ひニO i.e.,メ4=O. Therefore the desired assertion.N(K4)...1mod p follows at once from z4=O.([4],Theorem). LEMMA2.1∫ρ......一1modη,診h6πN(K2)......O mod2αη42〉(K2)...3modηhol4。 PROOF. Since(ρ一1,π)=1,we get#{α∈F1=GF(p)1απ=α1=1. This lead to #{α∈F2;α露=α/-n,because F2contains,in our case,anπth primitive root of unity. Moreover,it is clear that#{(α,β)∈F2×F21β2一αη一α,β≠Ol≡O mod2。Hence we get the first part of the lemma as follows. N(K2)=1十#{(α,O)∈172×F2;αη=8} +#{(α,β)∈F2×F2;β≠0,β』α”一α}≡...1+η≡O mod2。 We will now prove the second assertion.Since F2conta三ns anπth primitive root of unity,we have #{(α,β)∈F2×F、;αη=β2+α,α≠0}...≡0:modπ. Therefore,because of#{β∈F2;β2......一α}=2,we get N(K2)=1+#/(O,β)∈F2×F21β2・=一α} +#/(α,β)∈F2×F2;α≠O,β2一α”一‘zl...≡1+2...3mo(iη. This completes the proof of the lemma. Now let us tum to the proof of Theorem2. PROOF of Theorem2。As is well known,the inequalitiesρ2十1-29ρ≦N(K2)≦ρ2 十1十29ρhold.(M。Eichler[1],P.306). Therefore Lemma l and the first part of Lemma21ead to 2V(K2)一ρ2+1+卿(形一〇,±2,±4,......,±29). Using the second part of Lemma2,we haveρ2十1十郷ρ......3modη. Therefore we can easily get解=2g because of our assumptionsρ≡...一l modπand n=29十1. Hence we have our assertion1〉(K2)ニρ2十1十2gρ. 3. Results. Let us now consider the question in §10nly in the case ofη=5 andπ=7.In fact,we can answer our question in the case ofπ=5in the affirmative as follows. THEOREM3.L8君F・=GF(ρ)68α方競8ρプ麹8方8140∫6肋7疏爾5漉ρ≠2.L8オ K-F(諾,ッ)68αhッPθプ81妙痂血n6オプ・n方614・η6プ.F48伽8づ6ツαn69微漉・ηプー諾5-8 (α≠0,α∈F).Z)8π・」のッhオh召61α55π蹴み8プげK。1∫ρ≡4mod5,診h8nh一(ρ+1)2 例4L(%)一1+2勿2+ρ2π4. PROOF.ApPlying Theorem l to4=1we haveノ〉1=N(K1);ρ十1. Moreover, apPlying Theorem2to9=2,we have1〉(K2)=ρ2十4ρ十1. So we get N2=ρ(ρ十3)/2 in view of the fact that the relation among1〉1,ハ72and N(K2)is given,by N(K2) ・=2〉、十21〉2. Therefore,by making use of the formula(1),we can easily obtain ごz1=0,ζz2=・2ρ, L(%)=1十2ρ%2十ρ2π4and h=L(1)=(メ》十1)2. Finally we will give an affirmative answer to our question in the case ofπ=7. THEOREM4.LαF=GF(カ)68α五n漉ρ7伽8方8140∫6hαプ観副5痂ρ≠2,Lα K=F(‘τ,‘y) 68αh‘yメ》8プ6114》擁c ∫診¢π6擁on 方614 0ηθプ F 46Lβn846ツ‘zn 6gz¢‘z擁on ツ2=‘じ7一α (α≠0,召∈F).Z)eπ・詑δッ勉h861α∫∫n%〃z6eプ・∫K.1∫ρ・三6mod7,彦h8nん一(ρ+1)3 4 Tadashi ~YVASHIO and L(u) = I + 3pu2 + 3p2u' + p3u6 . PROOF . As applications of Theorem I to d=1 and d=3, we have N*= N(K*) = p+1 and N(K3) =p3+1 . Consequently the formula N(K3) = N* + 3N3 Ieads to N*= (p3-p)/3 . Similarly , applying Theorem 2 to g = 3 , we have N(K,) = p' + 6p + I . Therefore the formula N(K2) =N*+2N2 also leads to N2= (p2+5p)/2. Hence, by means of the formula (1), rt Is easy to check on al=a3=0, aa=3p, L(u) =1+3pu2+3p'u'+p3u' and h=L(1) =(p+1)". This completes the proof of the theorem .

Frequency generator for generating signalling frequencies in a multifrequency telephone apparatus.

Abstract: of EP00212961. Frequency generator for generating signalling frequencies in a multifrequency telephone apparatus comprising a quartz oscillator (1) ; a first frequency divider (2) which is connected to the quartz oscillator (1) and has a fixed divisor ; a frequency divider (3, 4) which is connected to the first frequency divider (2) and has an integral divisor which can be set by a programming network (7) for generating frequencies selected through the programming network (7) ; and a digital/analog converter comprising a fourth frequency divider (5) with a fixed divisor, and a weight network (6) connected to the steps of the fourth frequency divider (5) for forming step-shaped signals of approximate sine wavefrom ; characterized in that the frequency divider (3, 4) with a settable integral divisor comprises and AND member (37) and an auxiliary flip-flop (4) as last step ; in that the output of the flip-flop (4h via 43h) is connected to a first input of the AND member (37) ; in that at least one further input of the AND member (37) is connected to the output of a divider step (31, 32) preceding the auxiliary flip-flop (4h) ; and in that the output of the AND member (37 via 341) is connected to the reset input of the auxiliary flip-flop (4h), and, controlled by the programming network (7 via 731, 732, 741, ..., 74p), to the set or reset input of at least one divider step (31, 32, 41, ..., 4p) preceding the auxiliary flip-flop (4h).