Пентаци

Пентаци, пиллĕкмĕш ретри гипероператор. Чăн та, ак çапла пулса тухать.

1. Хушасси, 1-мĕш ретри гипероператор:

   $a+n=a+\underbrace {1+1+\cdots +1} _{n}$

2. хутласси, 2-мĕш ретри гипероператор:

   $a\times n=\underbrace {a+a+\cdots +a} _{n}$

3. капаша хăпартасси, 3-мĕш ретри гипероператор:

   $a^{n}=\underbrace {a\times a\times \cdots \times a} _{n}$

4. тетраци, 4-мĕш ретри гипероператор:

   ${^{n}a}=\underbrace {a^{a^{\cdot ^{\cdot ^{a}}}}} _{n}$

5. пентаци, 5-мĕш ретри гипероператор:

   ${_{n}a}=\underbrace {^{^{^{^{a}\cdot }\cdot }a}a} _{n}$

Кашни черетлĕ операци умĕнхин итерацийĕ пулса тăрать.

Çăлкуçсем

• Perstein, Millard H. (June 1961), "Algorithm 93: General Order Arithmetic", Communications of the ACM, 5 (6): 344, doi:10.1145/367766.368160, S2CID 581764.
• Goodstein, R. L. (1947), "Transfinite ordinals in recursive number theory", The Journal of Symbolic Logic, 12 (4): 123–129, doi:10.2307/2266486, JSTOR 2266486, MR 0022537, S2CID 1318943.
• Knuth, D. E. (1976), "Mathematics and computer science: Coping with finiteness", Science, 194 (4271): 1235–1242, Bibcode:1976Sci...194.1235K, doi:10.1126/science.194.4271.1235, PMID 17797067, S2CID 1690489.
• Blakley, G. R.; Borosh, I. (1979), "Knuth's iterated powers", Advances in Mathematics, 34 (2): 109–136, doi:10.1016/0001-8708(79)90052-5, MR 0549780.
• Conway, John Horton; Guy, Richard (1996), The Book of Numbers, Springer, p. 61, ISBN 9780387979939.
• "Tetration.org - Tetration". www.tetration.org. Retrieved 2022-09-12.
• Nambiar, K. K. (1995), "Ackermann functions and transfinite ordinals", Applied Mathematics Letters, 8 (6): 51–53, CiteSeerX 10.1.1.563.4668, doi:10.1016/0893-9659(95)00084-4, MR 1368037.
• Paulsen, W.; Cowgill, S. (March 2017). "Solving {\displaystyle F(z+1)=b^{F(z)}} {\displaystyle F(z+1)=b^{F(z)}} in the complex plane" (PDF). Advances in Computational Mathematics. 43: 1–22. doi:10.1007/s10444-017-9524-1. S2CID 9402035.
• Kneser, H. (1950). "Reelle analytische Lösungen der Gleichung {\displaystyle \varphi {\Big (}\varphi (x){\Big )}={\rm {e}}^{x}} {\displaystyle \varphi {\Big (}\varphi (x){\Big )}={\rm {e}}^{x}} und verwandter Funktionalgleichungen". Journal für die reine und angewandte Mathematik (in German). 187: 56–67.