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%I #13 Oct 20 2019 01:57:27
%S 5,17,23,37,47,53,61,79,83,97,101
%N Defiant primes of order 3. Primes p such that no prime numerator and denominator of the continued fraction rational approximation of sqrt(p) exist for numerators less than 10^3 digits in length.
%C Defiant primes of order k are also of order r where 0 < r < k.
%e The 8th convergent of sqrt(5) is c = 51841/23184. c^2 = 5.00000000186... but both numerator and denominator are nonprime.
%o (PARI) cfracnumdenomprime(m,f) = { default(realprecision,3000); cf = vector(m+10); x=f; for(n=0,m, i=floor(x); x=1/(x-i); cf[n+1] = i; ); for(m1=0,m, r=cf[m1+1]; forstep(n=m1,1,-1, r = 1/r; r+=cf[n]; ); numer=numerator(r); denom=denominator(r); if(ispseudoprime(numer)&&ispseudoprime(denom),print1(numer",");numer2=numer;denom2=denom); if(length(Str(numer))>999,break); ) }
%K frac,nonn,base
%O 1,1
%A _Cino Hilliard_, Jan 16 2005
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