Question

CMR: \(a^p\equiv a\)(mod p) với p là số nguyên tố, a là số nguyên (Định lý nhỏ Fermat)

Answer 1

ap&#x2212;1&#x2261;1(modp)&lt;=&gt;ap&#x2212;1&#x2212;1&#x22EE;p&lt;=&gt;ap&#x2212;a&#x22EE;p" role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-size:16.38px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; white-space:nowrap; word-spacing:normal" class="MathJax_CHTML mjx-chtml">ap−1≡1(modp)<=>ap−1−1⋮p<=>ap−a⋮p  (1)

np&#x2212;n&#x22EE;p" role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-size:16.38px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; white-space:nowrap; word-spacing:normal" class="MathJax_CHTML mjx-chtml">np−n⋮p

(n+1)p&#x2212;(n+1)=np+np&#x2212;1+n(n&#x2212;1)2!np&#x2212;2+...+n(n&#x2212;1)2!n2+n+1" role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-size:16.38px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; white-space:nowrap; word-spacing:normal" class="MathJax_CHTML mjx-chtml">(n+1)p−(n+1)=np+np−1+n(n−1)2!np−2+...+n(n−1)2!n2+n+1

Ckp=p(p&#x2212;1)...(p&#x2212;k+1)k!" role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-size:16.38px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; white-space:nowrap; word-spacing:normal" class="MathJax_CHTML mjx-chtml">Ckp=p(p−1)...(p−k+1)k!

(p&#x2212;1)...(p&#x2212;k+1)k!" role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-size:16.38px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; white-space:nowrap; word-spacing:normal" class="MathJax_CHTML mjx-chtml">(p−1)...(p−k+1)k!  là số nguyên và np−k cũng là số nguyên nên:

p(np&#x2212;1+p&#x2212;12!.np&#x2212;2+...+n)" role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-size:16.38px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; white-space:nowrap; word-spacing:normal" class="MathJax_CHTML mjx-chtml">p(np−1+p−12!.np−2+...+n) là số nguyên chia hết cho p.

(n+1)p&#x2212;n&#x2212;1=np+pm+1&#x2212;n&#x2212;1" role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-size:16.38px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; word-spacing:normal" class="MathJax_CHTML mjx-chtml">(n+1)p−n−1=np+pm+1−n−1(với m thuộc Z nào đó)

=np&#x2212;n+pm" role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-size:16.38px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; white-space:nowrap; word-spacing:normal" class="MathJax_CHTML mjx-chtml">=np−n+pm (dễ dàng thấy nó chia hết cho p)

ap&#x2212;a=&#x2212;bp+b=&#x2212;(bp&#x2212;b)&#x22EE;p" role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-size:16.38px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; white-space:nowrap; word-spacing:normal" class="MathJax_CHTML mjx-chtml">ap−a=−bp+b=−(bp−b)⋮p (với b là số nguyên dương, a=−b)

ap&#x2212;a&#x22EE;p" role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-size:16.38px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; white-space:nowrap; word-spacing:normal" class="MathJax_CHTML mjx-chtml">ap−a⋮p với mọi a∈Z

Bài viết đã được chỉnh sửa nội dung bởi Namthemaster1234: 08-07-2014 - 08:48