meditationatae

Just another WordPress.com site

testing Erdos-Alaoglu formula, 1944

p  = 1919563670117
? epsilon=log(1+1/p)/log(p)
%3412 = 1.841917512375253387582800136632078607012932929350472081631215817120\

0850387536075366360000346413577752892872978581781089040610273407756640119093\

566970912124987390341597763792633875533561968232945742411 E-14
? (p^(1+epsilon) -1)/(p^epsilon-1)-p^2
%3413 = -2.0632267959132986417 E-188

so,  (p^(1+epsilon) -1)/(p^epsilon-1) is very nearly  p^2 .

In:

On highly composite and similar numbers

from:

http://www.renyi.hu/~p_erdos/1944-03.pdf

Theorem 10, page 455,

with q=1919563670117 , epsilon as above,

(q^(1+epsilon) -1)/(q^epsilon-1)  is very nearly  q^2, from which

log{ (q^(1+epsilon) -1)/(q^epsilon-1) }/log(q) ~=  log{ q^2 }/log(q) = 2, and

k_q (epsilon) :=

floor( log{ (q^(1+epsilon) -1)/(q^epsilon-1) }/log(q)  )  -1  ~=  1 .

This is essentially formula   2-3  from Briggs [2006] :

“Abundant Numbers and the Riemann Hypothesis”

My formula for the critical epsilon where p (or q) takes the value 1,

epsilon=log(1+1/p)/log(p),

is based on case A of Section 3.1 of Briggs, i.e.

epsilon_ext := log_p ( 1+p)

which seems like it ought to be

epsilon_ext := log_p ( 1+ 1/p) …

for q=2 and the same epsilon,

? log((2^(1+epsilon) -1)/(2^epsilon-1))/log(2) -1
%3417 = 45.1545512475130379736096181270110144317384863\

86083957754338772387470581021435731002970125783467511\

725552566840042003987259913150851579042665075675624535\

698477597700295403619793870961792211397414207609

so k_2(epsilon) = 45, and one more:

q=1936559

log((1936559^(1+epsilon) -1)/(1936559^epsilon-1))/log(1936559) -1
%3419 = 2.0000010912692301796087068429724058514864454323011131921297632152332075037526434914883\

53398454193238741239518429029909352278247886113926444869697714205744612396290544880902591466952\

6646932563696752063

so k_1936559 (epsilon) = 2   whereas

for q = 1936579

? log((1936579^(1+epsilon) -1)/(1936579^epsilon-1))/log(1936579) -1
time = 1 ms.
%3421 = 1.9999996151807785320951198440160504978399381098524199172\

21987303184911001730509027273083098129918438554971909812373985563\

1307921566533923117160465527695430475266105852265898843702136254\

973097735862412

so that  k_1936579 (epsilon) = 1.

Advertisements

Written by meditationatae

December 23, 2014 at 10:36 am

Posted in History

%d bloggers like this: