Questions in category: 初等数论 (Elementary Number Theory)
数论 >> 一般数论 >> 初等数论
<[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] >

1. 关于 $10^n+1$ 的因子分解

Posted by haifeng on 2020-11-15 10:10:55 last update 2020-11-15 10:12:56 | Answers (1) | 收藏


关于 $10^n+1$ 的因子分解

 

2. 列出这样一些特殊的素数, $p=d_1d_2\ldots d_n$ 是素数, 它的反转(reverse) $q=\mathrm{reverse}(p)=d_n d_{n-1}\ldots d_2 d_1$ 也是素数.

Posted by haifeng on 2020-09-01 20:57:30 last update 2020-09-01 21:52:10 | Answers (0) | 收藏


列出这样一些特殊的素数, $p=d_1d_2\ldots d_n$ 是素数, 它的反转(reverse) $q=\mathrm{reverse}(p)=d_n d_{n-1}\ldots d_2 d_1$ 也是素数.

比如 91121 和 12119

 


使用 Calculator 进行验证.

>> isprime(91121)
in> isprime(91121)
This is function isprime
sqrtnum=301.862552
--------------------
we choose sqrtnum : 303
 ... since we use the traditional algorithm, ...
 ... please wait a minute ...
 ... ::: ... ::: ...
out> 91121 is a prime

------------------------

>> reverse(91121)
in> reverse(91121)
out> 12119

 

>> isprime(12119)
in> isprime(12119)
This is function isprime
sqrtnum=110.086330
--------------------
we choose sqrtnum : 111
 ... since we use the traditional algorithm, ...
 ... please wait a minute ...
 ... ::: ... ::: ...
out> 12119 is a prime

 

3. 连分数 $1+1/(2+1/(3+1/(4+1/(5+\cdots+))))$

Posted by haifeng on 2020-07-29 16:23:52 last update 2020-07-29 16:31:50 | Answers (0) | 收藏


(以下使用 Calculator 计算)

>> setprecision(200)
in> setprecision(200)
Now the precision is: 200

------------------------

>> continued_fraction(1,2,...,100)
in> continued_fraction(1,2,...,100)
out> 210628326606030925509504995396115751215082237173770949560136689651065020446811215535403721603460152269102594085067245222090841983806373539433329769060736986251|146971108555054591263734471078160507352050045572198741030186489794108431298591736983812285328810821610957278981960240168856854819813194627616076643581220741550

Expression:
1+1/(2+1/(3+1/(4+1/(5+1/(6+1/(7+1/(8+1/(9+1/(10+1/(11+1/(12+1/(13+1/(14+1/(15+1/(16+1/(17+1/(18+1/(19+1/(20+1/(21+1/(22+1/(23+1/(24+1/(25+1/(26+1/(27+1/(28+1/(29+1/(30+1/(31+1/(32+1/(33+1/(34+1/(35+1/(36+1/(37+1/(38+1/(39+1/(40+1/(41+1/(42+1/(43+1/(44+1/(45+1/(46+1/(47+1/(48+1/(49+1/(50+1/(51+1/(52+1/(53+1/(54+1/(55+1/(56+1/(57+1/(58+1/(59+1/(60+1/(61+1/(62+1/(63+1/(64+1/(65+1/(66+1/(67+1/(68+1/(69+1/(70+1/(71+1/(72+1/(73+1/(74+1/(75+1/(76+1/(77+1/(78+1/(79+1/(80+1/(81+1/(82+1/(83+1/(84+1/(85+1/(86+1/(87+1/(88+1/(89+1/(90+1/(91+1/(92+1/(93+1/(94+1/(95+1/(96+1/(97+1/(98+1/(99+1/(100)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))

TeX Code:
1+\frac{1}{2+\frac{1}{3+\frac{1}{4+\frac{1}{5+\frac{1}{6+\frac{1}{7+\frac{1}{8+\frac{1}{9+\frac{1}{10+\frac{1}{11+\frac{1}{12+\frac{1}{13+\frac{1}{14+\frac{1}{15+\frac{1}{16+\frac{1}{17+\frac{1}{18+\frac{1}{19+\frac{1}{20+\frac{1}{21+\frac{1}{22+\frac{1}{23+\frac{1}{24+\frac{1}{25+\frac{1}{26+\frac{1}{27+\frac{1}{28+\frac{1}{29+\frac{1}{30+\frac{1}{31+\frac{1}{32+\frac{1}{33+\frac{1}{34+\frac{1}{35+\frac{1}{36+\frac{1}{37+\frac{1}{38+\frac{1}{39+\frac{1}{40+\frac{1}{41+\frac{1}{42+\frac{1}{43+\frac{1}{44+\frac{1}{45+\frac{1}{46+\frac{1}{47+\frac{1}{48+\frac{1}{49+\frac{1}{50+\frac{1}{51+\frac{1}{52+\frac{1}{53+\frac{1}{54+\frac{1}{55+\frac{1}{56+\frac{1}{57+\frac{1}{58+\frac{1}{59+\frac{1}{60+\frac{1}{61+\frac{1}{62+\frac{1}{63+\frac{1}{64+\frac{1}{65+\frac{1}{66+\frac{1}{67+\frac{1}{68+\frac{1}{69+\frac{1}{70+\frac{1}{71+\frac{1}{72+\frac{1}{73+\frac{1}{74+\frac{1}{75+\frac{1}{76+\frac{1}{77+\frac{1}{78+\frac{1}{79+\frac{1}{80+\frac{1}{81+\frac{1}{82+\frac{1}{83+\frac{1}{84+\frac{1}{85+\frac{1}{86+\frac{1}{87+\frac{1}{88+\frac{1}{89+\frac{1}{90+\frac{1}{91+\frac{1}{92+\frac{1}{93+\frac{1}{94+\frac{1}{95+\frac{1}{96+\frac{1}{97+\frac{1}{98+\frac{1}{99+\frac{1}{100}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}

------------------------

>> 210628326606030925509504995396115751215082237173770949560136689651065020446811215535403721603460152269102594085067245222090841983806373539433329769060736986251/146971108555054591263734471078160507352050045572198741030186489794108431298591736983812285328810821610957278981960240168856854819813194627616076643581220741550
in> 210628326606030925509504995396115751215082237173770949560136689651065020446811215535403721603460152269102594085067245222090841983806373539433329769060736986251/146971108555054591263734471078160507352050045572198741030186489794108431298591736983812285328810821610957278981960240168856854819813194627616076643581220741550

out> 1.43312742672231175831718345577599182043151276790598052343442863639430918325417290013650372643578611465950013404308853642953017708273894637360407321952533635247368315637151340965862626563444808561719787

------------------------

 


其他参考资料

https://math.stackexchange.com/questions/69519/closed-form-for-a-pair-of-continued-fractions

4. 设 $g|ab$, $g|cd$ 及 $g|(ac+bd)$, 证明: $g|ac$ 且 $g|bd$.

Posted by haifeng on 2020-01-06 08:37:06 last update 2020-01-06 08:37:06 | Answers (2) | 收藏


设 $g|ab$, $g|cd$ 及 $g|(ac+bd)$, 证明: $g|ac$ 且 $g|bd$.

 

5. 证明: $x^2+2y^2=203$ 无整数解.

Posted by haifeng on 2020-01-02 17:00:58 last update 2020-01-02 17:00:58 | Answers (1) | 收藏


证明: $x^2+2y^2=203$ 无整数解.

 

 

References:

潘承洞, 潘承彪, 《数论》   P. 

6. 设 $N$ 不是平方数, 证明 $\sqrt{N}$ 是无理数.

Posted by haifeng on 2019-11-24 11:37:01 last update 2019-11-24 11:37:01 | Answers (0) | 收藏


设 $N$ 不是平方数, 证明 $\sqrt{N}$ 是无理数.

事实上, 只要 $N$ 不是某个整数 $n$ 的 $m$ 次幂, 则 $\sqrtn[m]{N}$ 都是无理数.

 

现在, 若 $\sqrt{N}$ 是无理数, 问是否有无理性的几何证明?

 

哈代数论中给出了 $\sqrt{5}$ 无理性的几何证明.

7. 用初等方法证明以下恒等式

Posted by haifeng on 2019-11-24 07:19:02 last update 2019-11-24 07:19:02 | Answers (0) | 收藏


\[
\begin{split}
&(1+2x+2x^4+\cdots)^6\\
=&1+16\biggl(\frac{1^2 x}{1+x^2}+\frac{2^2 x^2}{1+x^4}+\frac{3^2 x^3}{1+x^6}+\cdots\biggr)-4\biggl(\frac{1^2 x}{1-x}-\frac{3^2 x^3}{1-x^3}+\frac{5^2 x^5}{1-x^5}-\cdots\biggr)
\end{split}
\]

 

\[
(1+2x+2x^4+\cdots)^8=1+16\biggl(\frac{1^3 x}{1+x}+\frac{2^3 x^2}{1-x^2}+\frac{3^3 x^3}{1+x^3}+\cdots\biggr)
\]

 

References:

哈代数论(第6版)20.13 用多个平方和表示数

G. H. Hardy, E. M. Wright, An Introduction to the Theory of Numbers.
 

8. 证明: $30|(6n^5+15n^4+10n^3-n)$.

Posted by haifeng on 2019-11-04 22:01:29 last update 2019-11-04 22:01:29 | Answers (1) | 收藏


证明: $30|(6n^5+15n^4+10n^3-n)$.

9. 证明: $9|(n^3+(n+1)^3+(n+2)^3)$.

Posted by haifeng on 2019-11-04 21:28:05 last update 2019-11-04 21:28:05 | Answers (1) | 收藏


证明: $9|(n^3+(n+1)^3+(n+2)^3)$.

10. 3可以写成三个整数的立方和

Posted by haifeng on 2019-09-23 21:12:43 last update 2019-09-23 21:12:43 | Answers (0) | 收藏


569936821221962380720^3 + (-569936821113563493509)^3 + (-472715493453327032)^3 == 3

 

References:

数论群

http://bristol.ac.uk/maths/news/2019/number-3.html

<[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] >