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%I #69 Apr 05 2024 10:04:33
%S 7,9,11,12,34,46,49,65,89,95,103,127,144,150,163,172,206,236,249,258,
%T 275,288,300,309,312,385,492,495,505,577,641,683,729,738,751,796,835,
%U 873,904,990,995,1010,1154,1210,1297,1312,1403,1458,1476,1502,1544,1626,1661,1731,1808,1852,1985,1988,2020,2059,2107,2214,2304,2316,2370,2448,2594,2833,2840,2883,2920,3073,3088,3097
%N Positive integers z such that pi(x^3+y^3) = pi(z^3) for some 0 < x <= y < z, where pi(m) denotes the number of primes not exceeding m.
%C The sequence has infinitely many terms. In fact, for any integer t > 0, we have (6*t^2)^3 + (6*t^3-1)^3 = (6*t^3+1)^3 - 2 and hence pi((6*t^2)^3+(6*t^3-1)^3) = pi((6*t^3+1)^3) since neither (6*t^3+1)^3 nor (6*t^3+1)^3-1 is prime.
%C Concerning the equation pi(x^3+y^3) = pi(z^3) with 0 < x <= y < z, there are exactly 70 solutions with z <= 2700. They are (x,y,z) = (5,6,7),(6,8,9),(7,10,11),(9,10,12),(15,33,34),(23,44,46),(24,47,49),(43,58,65),(41,86,89),(47,91,95),(64,94,103),(95,106,127),(71,138,144),(73,144,150),(54,161,163),(135,138,172),(128,188,206),(55,235,236),(135,235,249),(197,212,258),(159,256,275),(142,276,288),(146,288,300),(192,282,309),(161,297,312),(96,383,385),(252,345,385),(390,391,492),(334,438,495),(372,426,505),(426,486,577),(297,619,641),(353,650,683),(242,720,729),(244,729,738),(150,749,751),(602,659,796),(161,833,835),(470,825,873),(566,823,904),(668,876,990),(514,947,995),(744,852,1010),(791,812,1010),(509,1120,1154),(852,972,1154),(236,1207,1210),(216,1295,1297),(459,1293,1312),(915,1259,1403),(484,1440,1458),(488,1458,1476),(300,1498,1502),(368,1537,1544),(511,1609,1626),(420,1652,1661),(1278,1458,1731),(1132,1646,1808),(1033,1738,1852),(1241,1808,1985),(1010,1897,1988),(1582,1624,2020),(294,2057,2059),(237,2106,2107),(732,2187,2214),(575,2292,2304),(577,2304,2316),(1518,2141,2370),(1611,2189,2448),(432,2590,2594).
%C Recall Fermat's Last Theorem, which asserts that the Diophantine equation x^n + y^n = z^n with n > 2 and x,y,z > 0 has no solution. In 1936 K. Mahler discovered that
%C (9*t^3+1)^3 + (9*t^4)^3 - (9*t^4+3*t)^3 = 1.
%C Conjecture: (i) For any integers n > 3 and x,y,z > 0 with {x,y} not equal to {1,z}, we have |x^n+y^n-z^n| >= 2^n-2, unless n = 5, {x,y} = {13,16} and z = 17.
%C (ii) For any integer n > 3 and x,y,z > 0 with {x,y} not containing z, there is a prime p with x^n+y^n < p < z^n or z^n < p < x^n+y^n, unless n = 5, {x,y} = {13,16} and z = 17.
%C (iii) For any integers n > 3, x > y >= 0 and z > 0 with x not equal to z, there always exists a prime p with x^n-y^n < p < z^n or z^n < p < x^n-y^n.
%C We have verified part (i) of the conjecture for n = 4..10 and 0 < x,y,z <= 1700.
%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 Chai Wah Wu, <a href="/A262536/b262536.txt">Table of n, a(n) for n = 1..446</a>
%H M. Beck, E. Pine, W. Tarrant and K. Y. Jensen, <a href="https://doi.org/10.1090/S0025-5718-07-01947-3">New integer representations as the sum of three cubes</a>, Math. Comp. 76(2007), 1683-1690.
%H V. L. Gardiner, R. B. Lazarus, P. R. Stein, <a href="http://dx.doi.org/10.1090/S0025-5718-1964-0175843-9">Solutions of the diophantine equation x^3+y^3=z^3-d</a>, Math. Comp. 18 (1964) 408-413
%H K. Mahler, <a href="https://carmamaths.org/resources/mahler/docs/040.pdf">Note on hypothesis K of Hardy and Littlewood</a>, J. London Math. Soc. 11(1936), 136-138.
%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.
%H A. J. Wiles, <a href="http://math.stanford.edu/~lekheng/flt/wiles-small.pdf">Modular elliptic curves and Fermat's last theorem</a>, Ann. Math. 141 (1995), 443-551.
%e a(1) = 7 since pi(5^3+6^3) = pi(125+216) = pi(341) = 68 = pi(343) = pi(7^3).
%e a(2) = 9 since pi(6^3+8^3) = pi(216+512) = pi(728) = 129 = pi(729) = pi(9^3).
%e a(50) = 1502 since pi(300^3+1498^3) = pi(27000000+3361517992) = pi(3388517992) = 162202081 = pi(3388518008) = pi(1502^3).
%t pi[n_]:=PrimePi[n]
%t n=0;Do[Do[If[pi[x^3+y^3]==pi[z^3],n=n+1;Print[n," ",z];Goto[aa]],{x,1,z-1},{y,x,z-1}];Label[aa];Continue,{z,1,2700}]
%Y Cf. A000578, A000720, A019590, A262408, A262409, A262443, A262462.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Sep 24 2015
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