查看“︁超質數”︁的源代码

{{NoteTA
|G1 = Math
}}
'''超質數'''也稱為'''高階質數'''，是指在[[質數]]序列中，第2個、第3個、第5個……等序數為[[質數]]的數。換句話說，若將正整數和質數從小到大兩兩對應排列，讓正整數的1對應質數的2，則正整數那列為質數的數字，質數那列對應的就是超質數。

超質數有
:3, 5, 11, 17, 31, 41, 59, 67, 83, 109, 127, 157, 179, 191, 211, 241, 277, 283, 331, 353, 367, 401, 431, 461, 509, 547, 563, 587, 599, 617, 709, 739, 773, 797, 859, 877, 919, 967, 991, ... {{OEIS|id=A006450}}.

若''p''(''i'') 表示第''i''個質數，則超質數即為''p''(''p''(''i''))。
{| class="wikitable" style="text-align: center;"
|-
! ''n''
| 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16 || 17 || 18 || 19 || 20
|-
! ''p''(''n'')
| 2 || 3 || 5 || 7 || 11 || 13 || 17 || 19 || 23 || 29 || 31 || 37 || 41 || 43 || 47 || 53 || 59 || 61 || 67 || 71
|-
! ''p''(''p''(''n''))
| 3 || 5 || 11 || 17 || 31 || 41 || 59 || 67 || 83 || 109 || 127 || 157 || 179 || 191 || 211 || 241 || 277 || 283 || 331 || 353
|}

{{harvtxt|Dressler|Parker|1975}}利用電腦輔助的證明（和[[子集和問題]]的計算有關）證明了所有大於96的數都可以表示為幾個相異超質數的和。此證明的基礎和[[伯特蘭-切比雪夫定理]]有關，說明（大於11的每一個超質數，都比前一個的二倍要小。

Broughan及Barnett<ref>Kevin A. Broughan and A. Ross Barnett, [http://www.cs.uwaterloo.ca/journals/JIS/VOL12/Broughan/broughan16.html On the Subsequence of Primes Having Prime Subscripts] {{Wayback|url=http://www.cs.uwaterloo.ca/journals/JIS/VOL12/Broughan/broughan16.html |date=20150911024105 }}, ''Journal of Integer Sequences'' '''12''' (2009), article 09.2.3.</ref>證明了小於x的超質數數量如下
:<math>\frac{x}{(\log x)^2}+O\left(\frac{x\log\log x}{(\log x)^3}\right)</math>

這可以說明超質數的集合是[[倒數和發散|小集]]（集合倒數的和會收斂）。

也可以用類似的方式定義更高階的質數，產生類似的數列{{harvtxt|Fernandez|1999}}。

超質數的一個變體是序數為[[回文素数]]的質數，數列如下
:3, 5, 11, 17, 31, 547, 739, 877, 1087, 1153, 2081, 2381, ... {{OEIS|id=A124173}}.

==參考資料==
<references/>
*{{citation | last1 = Bayless | first1 = Jonathan | last2 = Klyve | first2 = Dominic | last3 = Oliveira e Silva | first3 = Tomás | journal = Integers | mr = 3097157 | pages = A43:1–A43:21 | title = New bounds and computations on prime-indexed primes | volume = 13 | year = 2013 | url = http://digitalcommons.cwu.edu/cgi/viewcontent.cgi?article=1004&context=math | accessdate = 2024-03-11 | archive-date = 2023-06-07 | archive-url = https://web.archive.org/web/20230607164959/https://digitalcommons.cwu.edu/cgi/viewcontent.cgi?article=1004&context=math | dead-url = no }}
*{{citation
 | last1 = Broughan
 | first1 = Kevin A.
 | last2 = Barnett
 | first2 = A. Ross
 | journal = Journal of Integer Sequences
 | at = article 09.2.3
 | title = On the subsequence of primes having prime subscripts
 | url = http://www.cs.uwaterloo.ca/journals/JIS/VOL12/Broughan/broughan16.html
 | volume = 12
 | year = 2009
 | accessdate = 2015-08-25
 | archive-date = 2015-09-11
 | archive-url = https://web.archive.org/web/20150911024105/http://www.cs.uwaterloo.ca/journals/JIS/VOL12/Broughan/broughan16.html
 | dead-url = no
 }}
*{{citation
 | first1 = Robert E. | last1 = Dressler
 | first2 = S. Thomas | last2 = Parker
 | title = Primes with a prime subscript
 | journal = Journal of the ACM
 | volume = 22 
 | issue = 3 
 | year = 1975 
 | pages = 380–381
 | doi = 10.1145/321892.321900
 | mr = 0376599}}.
*{{citation
 | first1 = Neil
 | last1 = Fernandez
 | title = An order of primeness, F(p)
 | url = http://borve.org/primeness/FOP.html
 | year = 1999
 | accessdate = 2015-08-25
 | archive-date = 2012-07-10
 | archive-url = https://web.archive.org/web/20120710231820/http://www.borve.org/primeness/FOP.html
 | dead-url = no
 }}.

==外部連結==
*[https://web.archive.org/web/20071013121712/http://acm.sgu.ru/problem.php?contest=0&problem=116 A Russian programming contest problem related to the work of Dressler and Parker]

{{質數}}
[[Category:素數]]