3,184 research outputs found
On SA, CA, and GA numbers
Gronwall's function is defined for by where is the sum of the divisors of . We call an
integer a \emph{GA1 number} if is composite and for
all prime factors of . We say that is a \emph{GA2 number} if for all multiples of . In arXiv 1110.5078, we used Robin's
and Gronwall's theorems on to prove that the Riemann Hypothesis (RH) is
true if and only if 4 is the only number that is both GA1 and GA2. Here, we
study GA1 numbers and GA2 numbers separately. We compare them with
superabundant (SA) and colossally abundant (CA) numbers (first studied by
Ramanujan). We give algorithms for computing GA1 numbers; the smallest one with
more than two prime factors is 183783600, while the smallest odd one is
1058462574572984015114271643676625. We find nineteen GA2 numbers ,
and prove that a GA2 number exists if and only if RH is false, in
which case is even and .Comment: 29 pages, 2 tables, to appear in The Ramanujan Journal; added that
when Robin published his criterion for RH he was unaware of Ramanujan's
theore
A Randomized Sublinear Time Parallel GCD Algorithm for the EREW PRAM
We present a randomized parallel algorithm that computes the greatest common
divisor of two integers of n bits in length with probability 1-o(1) that takes
O(n loglog n / log n) expected time using n^{6+\epsilon} processors on the EREW
PRAM parallel model of computation. We believe this to be the first randomized
sublinear time algorithm on the EREW PRAM for this problem
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