Hello, I came across ’The Google Test’ by Eugene McDonnell (https://www.jsoftware.com/papers/play211.htm <https://www.jsoftware.com/papers/play211.htm>) and was wondering how to calculate many digits of Euler’s number e.
I found an (ancient) article "The calculation of e to many significant digits" by A. H. J. Sale in The Computer Journal 11(2) · August 1968 (https://www.researchgate.net/publication/266281843_The_Calculation_of_e_to_Many_Significant_Digits <https://www.researchgate.net/publication/266281843_The_Calculation_of_e_to_Many_Significant_Digits>) which uses a clever algorithm implemented in Algol 60 (!) to calculate e digit by digit and using only integer arithmetic. With this algorithm in J (see below) I could calculate as many digits of e as I want (I used 140 digits) and solved the Google Test which asks for: 1. First 10-digit prime found in consecutive digits of e. 2. The number F(5) which follows F(1)= 7182818284 F(2)= 8182845904 F(3)= 8747135266 F(4)= 7427466391 F(5)=__________ (BTW In the article of Eugene McDonell the second number F(2) contains transposed digits 9 and 0.) Here is my question: How would you calculate many significant digits of e in J? Regards, Ulrich NB. *** problem 1 *** NB. value to check m checkm =: 3 : 0 r =. -: ^. 6.2831852 * y r + y * (^. y) - 1 ) NB. given the number of digits of e which are wanted, NB. calculate the number of required terms number_of_terms =: 3 : 0 test =. 2.30258509 * >: y (>: ^: (test&>: @: checkm) ^:_) 4 ) NB. from JforC LoopWithInitial =: 2 : 'u&.>/\.&.(,&(<v))&.|.&.(<"_1)' NB. x is coeff and y is j, carry, follwed by the coeffs so far NB. answer are the next j and carry, the new coeff, and old coeffs term =: 4 : 0 j =. {. y carry =. 1 { y coeffs =. 2 }. y temp =. carry + 10 * x (<: j), ((0, j) #: temp), coeffs ) NB. initialise with m init =: 3 : 0 term LoopWithInitial (y, 0) (<: y) $ 1 ) NB. given is an output of term LoopWithInitial NB. find next output next =: 3 : 0 NB. take last row of output of term LoopWithInitial NB. the row contains: j carry coeffs last =. {: y NB. carry is the next digit of e, forget j ee =: ee , ": 1 { last term LoopWithInitial ((<: #last), 0) |. 2 }. last ) NB. calculate the number of digits of e NB. the digits are collected in a global variable ee calc_e =: 3 : 0 ee =: '' next^:y init number_of_terms y ee ) digits_e =: ". 10 ]\ calc_e 140 solution1 =: {. (I. 1 p: digits_e) { digits_e NB. -> 7427466391 NB. *** problem 2 *** NB. cross sum of a number csum =: 3 : '+/ "."0 ": y' NB. the given numbers have all the same cross sum of 49 NB. csum"0 (7182818284 8182845904 8747135266 7427466391) -> 49 49 49 49 NB. find the next number in the digits of e with a cross sum of 49 solution2 =: 4 }. (] {~ [: I. 49 = csum"0) digits_e NB. -> 5966290435 ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
