A B C D E F G H I J K L M N O P Q R S T U V W X Y Z All
Pillai, S. S.
- On ax - bY = bY ≥ax
Authors
1 Calcutta University, IN
Source
The Journal of the Indian Mathematical Society, Vol 8 (1944), Pagination: 10-13Abstract
In two previous papers, I have proved that the equation ax-by = c has only a finite number of solutions for any c, and has at most one solution when c is large. These two results together mean that ax-bY = ax - by has got only a finite number of solutions in x, y, X, T. The object of this paper is to prove that ax-bY = by &&8805; a.
- On the Smallest Primitive Root of a Prime
Authors
1 Calcutta University, IN
Source
The Journal of the Indian Mathematical Society, Vol 8 (1944), Pagination: 14-17Abstract
Let g(p) be the smallest primitive ischolar_main of a prime p. Vinogradowf has proved that g(p) = 0(p1/2+e) for every positive e. The object of this note is to prove that g(P) = Ω (log logp). (1)- Highly Composite Numbers of the tth Order
Authors
1 Calcutta University, IN
Source
The Journal of the Indian Mathematical Society, Vol 8 (1944), Pagination: 61-74Abstract
Let dt(N) denote the number of ways of decomposing JV into / factors. N is defined as a highly composite number of the th order, if df (N) > dt(N') for all N' < N. Ramanujan's highly composite numbers form the particular case when t = 2.- On Waring's Problem with Powers of Primes
Authors
1 Calcutta University, IN
Source
The Journal of the Indian Mathematical Society, Vol 8 (1944), Pagination: 18-20Abstract
Let pθ be the highest power of the prime p which divides the integer k, and let
γ{θ + 2 when p = 2 and 2|k
θ + 1 otherwise.
- On a Congruence Property of the Divisor Function
Authors
1 Calcutta, IN
Source
The Journal of the Indian Mathematical Society, Vol 6 (1942), Pagination: 118-119Abstract
Let d(n) denote the number of divisors of n, and N(k, x) the number of n≤x, for which d(n) is a multiple of k. The object of this paper is to obtain an asymptotic formula for N(k, x) when k is a prime.- On the Divisors of an+I
Authors
1 Calcutta, IN
Source
The Journal of the Indian Mathematical Society, Vol 6 (1942), Pagination: 120-121Abstract
Are there infinitely many primes which do not divide an+I for any n? The object of this note is to answer this question.- On Waring's Problem VIII
Authors
1 Annamalainagar, IN
Source
The Journal of the Indian Mathematical Society, Vol 3 (1939), Pagination: 205-220Abstract
Let G(f) denote the least value of s such that every large integer is the sum of s or fewer values of the polynomial f(x) of degree n. Loo-Keng Hua under certain conditions and the author under milder conditions independently proved certain results which are roughly equivalent to the result
G(f)=O(n3 log n).
In this paper under still milder conditions certain results about G(f) are proved, which are roughly equivalent to Vinogradow's result for G(n). A lemma of Hua (Lemma 12) plays an important part in this paper.
- On Waring's Problem IX
Authors
1 Annamalainagar, IN
Source
The Journal of the Indian Mathematical Society, Vol 3 (1939), Pagination: 221-225Abstract
Vinogradow has proved not only the amazing result that every large number is the sum of a finite number of nth powers of primes, but has also found out a formula for its upper bound. Let g1(n) denote the least value of s such that every number from the beginning is the sum of s nth powers of primes or unity. Then it is likely that when n > n0
g1(n)=2n+l-2,
where l=[(3/2)n].
The object of this paper is to prepare the way for it.
- A Note on the Paper of Sambasiva Rao
Authors
Source
The Journal of the Indian Mathematical Society, Vol 3 (1939), Pagination: 266-267Abstract
In the notation of the previous paper, Mr. Rao's results are sharpened to the following
THEOREM,
Max S k(x)=log logx/log{k/(k-1)} + M(k)+O(k log k),
where M(k)=Max Sk(x) when l ≤ x ≤ (2k)k; and the constant in O is independent of k and x.
- On Waring's Problem
Authors
1 Annamalai University, IN
Source
The Journal of the Indian Mathematical Society, Vol 2 (1937), Pagination: 16-44Abstract
Let g(n) denote the least value of s required to represent every positive integer as the sum of s non-negative nth powers. Further let l=[(3/2)n] and i=[(4/3)n], where [x] denotes the integral part of x.- On Sets of Square-Free Numbers
Authors
1 Annamalai University, IN
Source
The Journal of the Indian Mathematical Society, Vol 2 (1937), Pagination: 116-118Abstract
Let N(x)=N (x,d1,d2,.....,dr-1) denote the number of groups of square-free numbers q1,q2,...,qr not exceeding x, such that qm-q1=dm-1. (m=2, 3, . . ., r). Further let f(p) be the number of different residue classes modulus p2 and not congruent to 0 contained in the set d1,d2,.. dr-1.- On ax-by=c
Authors
1 Annamalai University, IN
Source
The Journal of the Indian Mathematical Society, Vol 2 (1937), Pagination: 119-122Abstract
Aaron Hersehefield has recently proved the very interesting result that
2x-3y=c (1)
has got at most one solution when c is large. Further, by applying the following theorem of Siegel, namely,
axn-byn=k (fixed n ≥ 3)
has at most one solution if |ab| is sufficiently large, he proves that
ax-by=c (2)
has at most 9 solutions.
- Waring's Problem V:On g(6)
Authors
1 Annamalainagar, IN
Source
The Journal of the Indian Mathematical Society, Vol 2 (1937), Pagination: 213-214Abstract
1. L. E. Dickson has proved that g(6)≤110. In this note I prove that
g(6)≤104.
Unless the base is indicated, the logarithm is taken to the base e.
2. In R (2) put s=13 and n=6.