A262536 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.

7, 9, 11, 12, 34, 46, 49, 65, 89, 95, 103, 127, 144, 150, 163, 172, 206, 236, 249, 258, 275, 288, 300, 309, 312, 385, 492, 495, 505, 577, 641, 683, 729, 738, 751, 796, 835, 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

Offset

1,1

Comments

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.

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).

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

(9*t^3+1)^3 + (9*t^4)^3 - (9*t^4+3*t)^3 = 1.

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.

(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.

(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.

We have verified part (i) of the conjecture for n = 4..10 and 0 < x,y,z <= 1700.

References

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.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..446

M. Beck, E. Pine, W. Tarrant and K. Y. Jensen, New integer representations as the sum of three cubes, Math. Comp. 76(2007), 1683-1690.

V. L. Gardiner, R. B. Lazarus, P. R. Stein, Solutions of the diophantine equation x^3+y^3=z^3-d, Math. Comp. 18 (1964) 408-413

K. Mahler, Note on hypothesis K of Hardy and Littlewood, J. London Math. Soc. 11(1936), 136-138.

Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

A. J. Wiles, Modular elliptic curves and Fermat's last theorem, Ann. Math. 141 (1995), 443-551.

Example

a(1) = 7 since pi(5^3+6^3) = pi(125+216) = pi(341) = 68 = pi(343) = pi(7^3).
a(2) = 9 since pi(6^3+8^3) = pi(216+512) = pi(728) = 129 = pi(729) = pi(9^3).
a(50) = 1502 since pi(300^3+1498^3) = pi(27000000+3361517992) = pi(3388517992) = 162202081 = pi(3388518008) = pi(1502^3).

Mathematica

pi[n_]:=PrimePi[n]
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}]

Crossrefs

Cf. A000578, A000720, A019590, A262408, A262409, A262443, A262462.

Sequence in context: A162308 A191883 A108815 * A161992 A334102 A167377

Adjacent sequences: A262533 A262534 A262535 * A262537 A262538 A262539

Keyword

nonn

Author

Zhi-Wei Sun, Sep 24 2015

Status

approved