级数与平方数

Treenewbee

{199,211,233,277,311,433,499,577,599,677,733,811,877,911,977,1777,1999,2111,2333,2777,2999,4111,4999,5333,7333,8111,8999,23333,47777,49999,59999,67777,79999,97777,199999,311111,511111,599999,611111,733333,799999,911111,2999999,4999999,19999999,29999999,59999999,61111111,71111111,83333333,89999999,577777777,799999999,1777777777,2777777777,8777777777,23333333333,31111111111,59999999999,67777777777,79999999999,311111111111,377777777777,2111111111111,5111111111111,17777777777777,31111111111111,41111111111111,47777777777777,53333333333333,59999999999999,67777777777777,499999999999999,577777777777777,1333333333333333,2777777777777777,5111111111111111,8777777777777777,23333333333333333,43333333333333333,377777777777777777,577777777777777777,1111111111111111111,2111111111111111111,2777777777777777777,5777777777777777777,29999999999999999999,89999999999999999999,97777777777777777777,911111111111111111111}

蔡家雄

不同于数论书中的小原根问题

若 3^n+2^(n+1) 是素数，

则 5 是素数 3^n+2^(n+1) 的原根。

简记为  \(g(3^n+2^{n+1})=5\) .

{{1,5},{2,5},{3,5},{4,5},{5,5},{6,5},{9,5},{11,5},{12,5},{15,5},{17,5},{22,5},{32,5},{33,5},{35,5},{36,5},{46,5},{47,5},{59,5},{63,5},{80,5},{101,5},{154,5},{159,5},{173,5}}


最小原根\(g=5\) 的质数公式，

—— \(g(3^{2n+1}+2^{2n+2})=5\) .

Treenewbee

蔡家雄 发表于 2023-3-7 10:41
若 8^(2n)+7^(2n+1) 是素数，则 10 是素数 8^(2n)+7^(2n+1) 的原根。

{2, 29}不符

蔡家雄

\(x^2 - 53*y^2 = ±1\) 的递推公式

\(x_{0}=1, x_{1}=182, x_{2}=66249, x_{n+1}=364*x_{n} + x_{n-1}\) ,

\(y_{0}=0, y_{1}=25, y_{2}=9100, y_{n+1}=364*y_{n} + y_{n-1}\) ,.

则 \(lim_{n→∞}\)   \(x_{n}/y_{n} = \sqrt{53}\) .

蔡家雄

\(x^2 - 61*y^2 = ±1\) 的递推公式

\(x_{0}=1, x_{1}=29718, x_{2}=1766319049, x_{n+1}=59436*x_{n} + x_{n-1}\) ,

\(y_{0}=0, y_{1}=3805, y_{2}=226153980, y_{n+1}=59436*y_{n} + y_{n-1}\) ,.

则 \(lim_{n→∞}\)   \(x_{n}/y_{n} = \sqrt{61}\) .

蔡家雄

\(x^2 - 73*y^2 = ±1\) 的递推公式

\(x_{0}=1, x_{1}=1068, x_{2}=2281249, x_{n+1}=2136*x_{n} + x_{n-1}\) ,

\(y_{0}=0, y_{1}=125, y_{2}=267000, y_{n+1}=2136*y_{n} + y_{n-1}\) ,.

则 \(lim_{n→∞}\)   \(x_{n}/y_{n} = \sqrt{73}\) .

蔡家雄

\(x^2 - 97*y^2 = ±1\) 的递推公式

\(x_{0}=1, x_{1}=5604, x_{2}=62809633, x_{n+1}=11208*x_{n} + x_{n-1}\) ,

\(y_{0}=0, y_{1}=569, y_{2}=6377352, y_{n+1}=11208*y_{n} + y_{n-1}\) ,.

则 \(lim_{n→∞}\)   \(x_{n}/y_{n} = \sqrt{97}\) .

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

Treenewbee

蔡家雄 发表于 2023-3-15 23:28
用公式法求解特殊佩尔方程

设 \(p=4k+1\) 是素数，

p=[5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449, 457, 461, 509, 521, 541, 557, 569, 577, 593, 601, 613, 617, 641, 653, 661, 673, 677, 701, 709, 733, 757, 761, 769, 773, 797, 809, 821, 829, 853, 857, 877, 881, 929, 937, 941, 953, 977, 997]

x^2 - p*y^2=1的最小解:
(5, [9, 4])
(13, [649, 180])
(17, [33, 8])
(29, [9801, 1820])
(37, [73, 12])
(41, [2049, 320])
(53, [66249, 9100])
(61, [1766319049, 226153980])
(73, [2281249, 267000])
(89, [500001, 53000])
(97, [62809633, 6377352])
(101, [201, 20])
(109, [158070671986249, 15140424455100])
(113, [1204353, 113296])
(137, [6083073, 519712])
(149, [25801741449, 2113761020])
(157, [46698728731849, 3726964292220])
(173, [2499849, 190060])
(181, [2469645423824185801, 183567298683461940])
(193, [6224323426849, 448036604040])
(197, [393, 28])
(229, [5848201, 386460])
(233, [1072400673, 70255304])
(241, [10085143557001249, 649641205044600])
(257, [513, 32])
(269, [13449, 820])
(277, [159150073798980475849, 9562401173878027020])
(281, [2262200630049, 134951575480])
(293, [12320649, 719780])
(313, [32188120829134849, 1819380158564160])
(317, [248678907849, 13967198980])
(337, [2063810353129713793, 112422913565764752])
(349, [169648201, 9081060])
(353, [10157115393, 540608704])
(373, [52387849, 2712540])
(389, [3287049, 166660])
(397, [838721786045180184649, 42094239791738433660])
(401, [801, 40])
(409, [25052977273092427986049, 1238789998647218582160])
(421, [3879474045914926879468217167061449, 189073995951839020880499780706260])
(433, [104564907854286695713, 5025068784834899736])
(449, [71798771299708449, 3388393513402120])
(457, [6983244756398928218113, 326662411570389853632])
(461, [1182351890184201, 55067617520620])
(509, [313201220822405001, 13882400040814700])
(521, [32961431500035201, 1444066532654320])
(541, [3707453360023867028800645599667005001, 159395869721270110077187138775196900])
(557, [27849, 1180])
(569, [16760473211643448449, 702635588524014320])
(577, [1153, 48])
(593, [721517598849, 29629176560])
(601, [38902815462492318420311478049, 1586878942101888360258625080])
(613, [464018873584078278910994299849, 18741545784831997880308784340])
(617, [3363593612801313, 135413180018248])
(641, [2609429220845977814049, 103066257550962737720])
(653, [10499986568677299849, 410896226494013260])
(661, [16421658242965910275055840472270471049, 638728478116949861246791167518480580])
(673, [4765506835465395993032041249, 183696788896587421699032600])
(677, [1353, 52])
(701, [277631049, 10485980])
(709, [665782673992201, 25003993164540])
(733, [195307849, 7213860])
(757, [3750107388553, 136299971388])
(761, [1280001, 46400])
(769, [535781868388881310859702308423201, 19320788325040337217824455505160])
(773, [3607394696649, 129748968980])
(797, [1221759532448649, 43276943002540])
(809, [376455160998025676163201, 13235458622462202510640])
(821, [9000987377460935993101449, 314136625452886403879740])
(829, [479835713751049, 16665383182260])
(853, [215454135724113414336120649, 7377009103065498851032020])
(857, [131822292741249, 4502963741200])
(877, [116476476553, 3933131148])
(881, [22606256615916825861249, 761624136944072910800])
(929, [13224937103288377430049, 433896111669844912840])
(937, [480644425002415999597113107233, 15701968936415353889062192632])
(941, [1068924905989944201, 34845956052079180])
(953, [15090531843660371073, 488830275367615376])
(977, [108832847723078562849, 3481871275306470280])
(997, [14418057673, 456624468])

飞石

平方公式延伸，
(a-1)平方=2a+1(a-2)平方等于2a-1……2a-3 2a-5.….2a-n
平方其实一个关于2的等差数列
例题:10的平方-8的平方=36
(10-1)*2+1=19,因为平方公差是2，所以九到八的差值是19-2=17
十减八的平方=19+17=36
用这种方法求解101的平方可以写成
一百的平方=10000 加(101对应的列表数值和)
101+100-201所以101的平方等于10000+201=100201
102的平方=10000+201+203=100404
例:5的平方=1+3+5+7+9=25
                     1, 3, 5, 7 ,9, 11…
                      1 4  9 16  25  36.…
立方的延伸公式关于一个6的等差数列
例2的立方-1的立方=(2-1)*6+1=7
10的立方-9的立方=271
10-1=9
9*6=54
6-54  =1 ，6，12，18，24，30，36，42，48，54
计算6到54等差数列的和，
由于一共有九个数，所以中间数是”5
对应数值等于5*6=30
所以九数和=（6+54)*4+30=270
1, 7,   19, 37,  61,    91,    127,    169,    217,     271
1 8    27  64  125  216    343     512      729    1000
1  6   12   18   24   30      36       42        48       54
由于第一位是1所以最后要加1。270+1=271
如果要计算10的立方-八的立方由于十对应的列表函数是54，
所以等于271-54=217
271+217=488就是十的立方减八的立方的差值
其中271对应的是10到九的数值
217对应的是九到八的数值。