数论小猜想

蔡家雄

佩尔方程有解定理

设 d 无 4k+3 的素因子，

且 d 仅有 2 和 8k+5 的素因子，

则 \(x^2 - d*y^2= -1\) 有正整数解。


佩尔方程无解定理

设 d 无 4k+3 的素因子，

且 d 有 2 和 8k+1 的素因子，

则 \(x^2 - d*y^2= -1\) 无正整数解。

cz1

方程 a^3+b^3=c^10 无解，

Treenewbee

cz1 发表于 2023-3-12 11:21
方程 a^3+b^3=c^10 无解，

\[8^3+8^3=2^{10}\]
\[729^3+1458^3=9^{10}\]
\[8192^3+8192^3=16^{10}\]

蔡家雄

若 5^(2k) -2 是素数，则 10 是素数 5^(2k) -2 的原根。

得 2k=2, 14, 26, 50, 126, 144, 260, 624, 1424, ......


若 5^(2k)+4 是素数，则 10 是素数 5^(2k)+4 的原根。

得 2k = 2，6，10，102，494，794，1326，5242, 5446, ......

蔡家雄

求 \(x^2 - (n^2 -2)*y^2=1\) 的最小解

则 \(x=n^2 -1 , y=n\) .

求 \(x^2 - (n^2+2)*y^2=1\) 的最小解

则 \(x=n^2+1 , y=n\) .

求 \(x^2 - ((2n)^2 -4)*y^2=1\) 的最小解

则 \(x=2*n^2 -1 , y=n\) .

求 \(x^2 - ((2n)^2+4)*y^2=1\) 的最小解

则 \(x=2*n^2+1 , y=n\) .

求 \(x^2 - ((2n+1)^2 -4)*y^2=1\) 的最小解

则 \(x=(2n+1)*(2n*(n+1) -1) , y=2n*(n+1)\) .

求 \(x^2 - ((2n+1)^2+4)*y^2=1\) 的最小解

则 \(x=2*((2n+1)^2+4)*(n^2+(n+1)^2) -1\) ,

     \(y=(4n+2)*(n^2+(n+1)^2)*(n^2+(n+1)^2+1)\) .

蔡家雄

求 \(x^2 - ((2n+1)^2+4)*y^2= -1\) 的最小解

则 \(x=(2n+1)*((2n+1)^2+3)/2\) , \(y=((2n+1)^2+1)/2\) .

蔡家雄

求 \(x^2 - ((2n+1)^2+4)*y^2= -1\) 的最小解

则 \(x=(2n+1)*((2n+1)^2+3)/2\) , \(y=((2n+1)^2+1)/2\) .

今天是国际数学日3月14日，推广此问题

求 \(x^2 - ((2n+1)^2+4^k)*y^2= -1\) 的最小解，

Treenewbee

蔡家雄 发表于 2023-3-14 20:37
若 (10t+5)^2+4 是素数，

则 10 是素数 (10t+5)^2+4 的原根。

t=1-10000,(10t+5)^2+4 是素数的t有2204个，其中不满足10 是素数 (10t+5)^2+4 的原根的有389个：
{21,27,37,45,61,118,136,147,175,188,216,237,253,273,283,292,305,328,346,364,394,496,535,559,571,574,595,607,679,723,726,756,760,769,805,820,823,832,862,875,879,883,901,907,908,979,1081,1105,1114,1261,1288,1336,1354,1393,1417,1428,1531,1599,1648,1654,1690,1707,1722,1738,1743,1744,1774,1784,1803,1828,1843,1882,1888,1891,1909,1917,1930,2015,2038,2050,2056,2075,2086,2104,2120,2135,2136,2170,2248,2257,2310,2316,2317,2337,2362,2368,2379,2383,2395,2422,2434,2440,2446,2455,2482,2497,2507,2512,2575,2650,2656,2708,2712,2744,2758,2782,2791,2833,2857,2862,2890,2939,2950,3046,3050,3065,3080,3118,3124,3133,3136,3157,3214,3262,3297,3409,3418,3427,3429,3444,3457,3472,3547,3598,3605,3626,3661,3667,3676,3685,3687,3721,3752,3758,3766,3804,3807,3817,3922,3967,4027,4036,4039,4051,4129,4153,4172,4186,4231,4237,4261,4285,4348,4402,4441,4444,4480,4498,4506,4522,4550,4564,4612,4703,4708,4718,4729,4732,4753,4781,4787,4858,4888,4891,4897,4942,4948,4956,4972,5013,5026,5035,5047,5056,5059,5060,5134,5167,5193,5209,5236,5308,5317,5383,5460,5485,5501,5511,5548,5563,5572,5749,5788,5824,5851,5863,5875,5936,5949,6046,6067,6096,6097,6109,6142,6146,6229,6290,6295,6307,6328,6337,6348,6370,6391,6397,6484,6493,6517,6534,6538,6569,6580,6589,6595,6638,6690,6721,6733,6738,6742,6769,6803,6826,6835,6964,6992,7060,7083,7126,7132,7135,7156,7198,7204,7216,7225,7234,7255,7258,7280,7321,7335,7381,7413,7427,7483,7549,7552,7566,7581,7617,7633,7666,7702,7711,7723,7744,7762,7765,7769,7774,7810,7895,7942,7972,8008,8020,8044,8071,8085,8116,8146,8155,8161,8167,8173,8215,8217,8254,8350,8372,8440,8461,8490,8515,8551,8563,8566,8590,8605,8636,8656,8668,8716,8731,8759,8764,8785,8788,8792,8800,8836,8841,8861,8869,8952,8963,8983,9008,9028,9036,9061,9082,9088,9145,9166,9187,9197,9232,9253,9283,9286,9316,9409,9435,9496,9514,9559,9637,9689,9710,9712,9724,9736,9745,9754,9778,9785,9787,9799,9858,9868,9883,9886,9954,9960,9970,9988}

蔡家雄

用公式法求解 x^2 - (4k+3)*y^2=1 的最小解，

要分 8k+3 与 8k+7 两种情形，在此不再讨论。

蔡家雄

用公式法求解特殊佩尔方程

设 \(p=4k+1\) 是素数，

求 \(x^2 - p*y^2=1\) 的最小解，

设 \(x=2p*r^2 -1\) , 求 最小的 \(r=?\) ,

使 \(y=((2p*r^2)*(2p*r^2 -2)/p)^{1/2}\) 是整数。

推论：此时，

设 \(p=4k+1\) 是素数，

求 \(x^2 - p*y^2= -1\) 的最小解，

得 \(y=r\) ,  \(x=((2p*r^2)*(2p*r^2 -2)/p)^{1/2}/(2*r)\) .