Problem 7:

Let  be the 20,000 x 20,000 matrix whose entries are zero everywhere except for the primes 2, 3, 5, 7, ..., 224737 along the main diagonal and the number 1 in all the positions  with ,2,4,8,...,16384.  What is the (1,1) entry of ?

Solution:

This is equivalent to computing  after solving , where . The system is diagonally dominant, and we use an iterative method: , where D is the diagonal of  (but without actually constructing the matrices). Gonnet used a  Darwin program, equivalent to the following in Maple.

 > b:= Array(1..20000,datatype=float[8]): b[1]:= 1: x:= Array(1..20000,datatype=float[8]): primes:= Array(1..20000,datatype=float[8]): for i from 1 to 20000 do primes[i]:= ithprime(i) od:

 > solvproc:= proc(N) local iter, toterr, i, rhs, ij, newxi; global x; for i from 1 to N do x[i]:= 0 od: for iter to 40 do     toterr := 0;     for i to N do     rhs := b[i];       ij := 1;     while ij < N do       if i-ij > 0 then rhs := rhs - x[i-ij] fi;       if i+ij <= N then rhs := rhs - x[i+ij] fi;       ij := 2*ij     od;     newxi := rhs/primes[i];     toterr := toterr + abs(newxi-x[i]);     x[i] := newxi   od;     if toterr=0 then break fi;     lprint(toterr); od: x[1]; end;

 > ti:= time(): evalhf(solvproc(20000)); (time()-ti)*seconds ;

`.887865995381123986`

`.278658134431981340`

`.873116873307076269e-1`

`.280984403047528780e-1`

`.896168412303717000e-2`

`.282597754444424655e-2`

`.885708717267995350e-3`

`.276700748083163562e-3`

`.862812417589902238e-4`

`.268719686136664988e-4`

`.836254516891092076e-5`

`.260111747308844605e-5`

`.808810146402529806e-6`

`.251448343535394878e-6`

`.781623535215165102e-7`

`.242947694034407002e-7`

`.755103880822366756e-8`

`.234685997925699352e-8`

`.729389779125172184e-9`

`.226686206265103620e-9`

`.704513858008347722e-10`

`.218954121769508914e-10`

`.680502535422857584e-11`

`.211472595988469877e-11`

`.656998092145587618e-12`

`.204068598900091197e-12`

`.635341005751694184e-13`

`.199326919187803528e-13`

`.616146405998221652e-14`

`.163608756217164570e-14`

`.409211670512742128e-15`

`.708330759345821984e-16`

`.934890384403199346e-19`

`.545716338525474910e-22`

`.355549011630070350e-29`

`.173686262298320443e-34`

This answer is correct to 14 digits.