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%I #43 Apr 14 2021 05:29:15
%S 1,2,4,24,41,51,88,95,99,179,183,663,782,829,1339,2054,2816,7918,8474,
%T 13264,16664,27415,39514,48606,51145,187222,200906,261980,353209,
%U 375162,396967,400469
%N Positive integers m such that pi(k^3) + pi(m^3) is a cube for some k = 1,...,m, where pi(x) denotes the number of primes not exceeding x.
%C Conjecture: (i) There are infinitely many distinct primes p,q,r such that pi(p^2) + pi(q^2) = r^2.
%C (ii) The Diophantine equation pi(x^3) + pi(y^3) = z^3 with 0 < x <= y and z >= 0 only has the following 17 solutions: (x,y,z) = (1,1,0), (2,2,2), (3,4,3), (16,24,13), (3,41,19), (37,51,26), (53,88,41), (18,95,41), (45,99,44), (108,179,79), (149,183,87), (8,663,251), (243,782,297), (803,829,385), (100,1339,489), (674,2054,745), (1519,2816,1047).
%C (iii) The Diophantine equation pi(x^n) + pi(y^n) = z^n with n > 3 and x,y,z > 0 has no solution.
%C a(26) > 10^5, if it exists. Conjecture (ii) above is false since these further solutions exist: (1339, 7918, 2682), (3360, 8474, 2922), (8443, 13264, 4764), (15590, 16664, 6696), (15883, 27415, 9431), (9719, 39514, 12689), (22265, 48606, 15933), (38606, 51145, 18297). - _Giovanni Resta_, Jun 14 2020
%C Further solutions: (79522, 187222, 58554), (65281, 200906, 61833), (222863, 261980, 92917), (226465, 353209, 114585), (41559, 375162, 112168), (244967, 396967, 127399), (291034, 400469, 133443) - _Chai Wah Wu_, Apr 13 2021
%D Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014-2016.
%e a(4) = 24 since pi(16^3) + pi(24^3) = pi(4096) + pi(13824) = 564 + 1633 = 2197 = 13^3.
%t f[n_]:=PrimePi[n^3]
%t CQ[n_]:=IntegerQ[n^(1/3)]
%t n=0;Do[Do[If[CQ[f[x]+f[y]],n=n+1;Print[n," ",y];Goto[aa]],{x,1,y}];Label[aa];Continue,{y,1,3000}]
%o (PARI) lista(nn) = {my(c=0, v=vector(nn)); for(m=1, nn, forprime(p=(m-1)^3+1, m^3, c++); v[m]=c; if(sum(k=1, m, ispower(v[k]+v[m], 3)), print1(m, ", "))); } \\ _Jinyuan Wang_, Jun 13 2020
%Y Cf. A000578, A000720, A019590, A262408, A262409, A262447, A262462, A262536.
%K nonn,more
%O 1,2
%A _Zhi-Wei Sun_, Sep 27 2015
%E a(18)-a(25) from _Giovanni Resta_, Jun 14 2020
%E a(26)-a(29) from _Chai Wah Wu_, Apr 05 2021
%E a(30)-a(32) from _Chai Wah Wu_, Apr 09 2021
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