## x^3 + y^3 = 13(z^3) and x^3 + y^3 = 31(z^3)

$x^3 \; + \; y^3 \; = \; 13 \, z^3$

$7^3 \; + \; 2^3 \; = \; 13 \; \times \; (3)^3$

$2513^3 \; + \; (-1388)^3 \; = \; 13 \; \times \; (1005)^3$

$26441619018689^3 \; + \; 40343602894936^3 \; = \; 13 \; \times \; (18636783082845)^3$

$x^3 \; + \; y^3 \; = \; 31 \, z^3$

$137^3 \; + \; (-65)^3 \; = \; 31 \; \times \; (42)^3$

$277028111^3 \; + \; 316425265^3 \; = \; 31 \; \times \; (119531076)^3$

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$(40208)^3 \; + \; (-22208)^3 \; = \; 13 \; \times \; (16080)^3$

$(203553)^3 \; + \; (-112428)^3 \; = \; 13 \; \times \; (81405)^3$

$(643328)^3 \; + \; (-355328)^3 \; = \; 13 \; \times \; (257280)^3$

$(2643958359322525696)^3 \; + \; (1732877944008802304)^3$
$= \; 13 \; \times \; (1221380216117329920)^3$

$(113565888938560867794944)^3 \; + \; (173274455036561044013056)^3$
$= \; 13 \; \times \; (80044373843465333637120)^3$

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math grad - Interest: Number theory
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### 2 Responses to x^3 + y^3 = 13(z^3) and x^3 + y^3 = 31(z^3)

1. paul says:

Here are a few of type x^3 + y^3 = 13 z^3.
Format is {x, y, z}

{2,7,3}
{4,14,6}
{6,21,9}
{8,28,12}
{10,35,15}
{12,42,18}
{14,49,21}
{16,56,24}
{18,63,27}
{20,70,30}
{22,77,33}
{24,84,36}
{26,91,39}
{28,98,42}

Paul.

• benvitalis says:

(7, 2, 3) is a primitive solution

(4, 14, 6) = 2 x [7, 2, 3 ]
(21, 6, 9) = 3 x [7, 2, 3 ]

(28, 8, 12) = 4 x [7, 2, 3 ]

(35, 10, 15) = 5 x [7, 2, 3]
(42, 12, 18) = 6 x [7, 2, 3 ]
(49, 14, 21) = 7 x [7, 2, 3 ]
(56, 16, 24) = 8 x [7, 2, 3 ]
(63, 18, 27) = 9 x [7, 2, 3 ]
(70, 20, 30) = 10 x [7, 2, 3 ]
(77, 22, 33) = 11 x [7, 2, 3 ]
(84, 24, 36) = 12 x [7, 2, 3 ]
(91, 26, 39) = 13 x [7, 2, 3 ]
(98, 28, 42) = 14 x [7, 2, 3 ]