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Artykuły w czasopismach na temat „Perfect numbers”

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Autor: Grafiati

Data publikacji: 4 czerwca 2021

Data aktualizacji: 25 stycznia 2023

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1

Holdener, Judy, i Emily Rachfal. "Perfect and Deficient Perfect Numbers". American Mathematical Monthly 126, nr 6 (29.05.2019): 541–46. http://dx.doi.org/10.1080/00029890.2019.1584515.

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2

Fu, Ruiqin, Hai Yang i Jing Wu. "The Perfect Numbers of Pell Number". Journal of Physics: Conference Series 1237 (czerwiec 2019): 022041. http://dx.doi.org/10.1088/1742-6596/1237/2/022041.

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3

Pollack, Paul, i Vladimir Shevelev. "On perfect and near-perfect numbers". Journal of Number Theory 132, nr 12 (grudzień 2012): 3037–46. http://dx.doi.org/10.1016/j.jnt.2012.06.008.

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4

Heath-Brown, D. R. "Odd perfect numbers". Mathematical Proceedings of the Cambridge Philosophical Society 115, nr 2 (marzec 1994): 191–96. http://dx.doi.org/10.1017/s0305004100072030.

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It is not known whether or not odd perfect numbers can exist. However it is known that there is no such number below 10300 (see Brent[1]). Moreover it has been proved by Hagis[4]and Chein[2] independently that an odd perfect number must have at least 8 prime factors. In fact results of this latter type can in priniciple be obtained solely by calculation, in view of the result of Pomerance[6] who showed that if N is an odd perfect number with at most k prime factors, thenPomerance's work was preceded by a theorem of Dickson[3]showing that there can be only a finite number of such N. Clearly however the above bound is vastly too large to be of any practical use. The principal object of the present paper is to sharpen the estimate (1). Indeed we shall handle odd ‘multiply perfect’ numbers in general, as did Kanold[5], who extended Dickson's work, and Pomerance. Our result is the following.

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5

Klurman, Oleksiy. "Radical of perfect numbers and perfect numbers among polynomial values". International Journal of Number Theory 12, nr 03 (23.03.2016): 585–91. http://dx.doi.org/10.1142/s1793042116500378.

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It is conjectured that [Formula: see text] for any perfect number [Formula: see text]. We prove that [Formula: see text] improving the previous bound of Luca and Pomerance as well as Acquaah and Konyagin. As a consequence, we prove that assuming the [Formula: see text]-conjecture, any integer polynomial of degree [Formula: see text] without repeated factors can take only finitely many perfect values. We also show that the latter holds unconditionally for even perfect numbers.

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6

Tang, Min, Xiao-Zhi Ren i Meng Li. "On near-perfect and deficient-perfect numbers". Colloquium Mathematicum 133, nr 2 (2013): 221–26. http://dx.doi.org/10.4064/cm133-2-8.

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7

Jiang, Xing-Wang. "On even perfect numbers". Colloquium Mathematicum 154, nr 1 (2018): 131–36. http://dx.doi.org/10.4064/cm7374-11-2017.

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8

Cowles, John, i Ruben Gamboa. "Perfect Numbers in ACL2". Electronic Proceedings in Theoretical Computer Science 192 (18.09.2015): 53–59. http://dx.doi.org/10.4204/eptcs.192.5.

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9

Bencze, Mihály. "About k-Perfect Numbers". Analele Universitatii "Ovidius" Constanta - Seria Matematica 22, nr 1 (10.12.2014): 45–50. http://dx.doi.org/10.2478/auom-2014-0005.

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10

Cohen, Peter, Katherine Cordwell, Alyssa Epstein, Chung-Hang Kwan, Adam Lott i Steven J. Miller. "On near-perfect numbers". Acta Arithmetica 194, nr 4 (2020): 341–66. http://dx.doi.org/10.4064/aa180821-11-10.

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11

CHEN, FENG-JUAN, i YONG-GAO CHEN. "ON ODD PERFECT NUMBERS". Bulletin of the Australian Mathematical Society 86, nr 3 (16.02.2012): 510–14. http://dx.doi.org/10.1017/s0004972712000032.

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AbstractLet q be an odd prime. In this paper, we prove that if N is an odd perfect number with qα∥N then σ(N/qα)/qα≠p,p2,p3,p4,p1p2,p21p2, where p,p1, p2 are primes and p1≠p2. This improves a result of Dris and Luca [‘A note on odd perfect numbers’, arXiv:1103.1437v3 [math.NT]]: σ(N/qα)/qα≠1,2,3,4,5. Furthermore, we prove that for K≥1 , if N is an odd perfect number with qα ∥N and σ(N/qα)/qα ≤K, then N≤4K8.

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12

TANG, MIN, i MIN FENG. "ON DEFICIENT-PERFECT NUMBERS". Bulletin of the Australian Mathematical Society 90, nr 2 (23.05.2014): 186–94. http://dx.doi.org/10.1017/s0004972714000082.

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AbstractFor a positive integer $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}n$, let $\sigma (n)$ denote the sum of the positive divisors of $n$. Let $d$ be a proper divisor of $n$. We call $n$ a deficient-perfect number if $\sigma (n) = 2n - d$. In this paper, we show that there are no odd deficient-perfect numbers with three distinct prime divisors.

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13

Finch, Carrie, i Lenny Jones. "Perfect power Riesel numbers". Journal of Number Theory 150 (maj 2015): 41–46. http://dx.doi.org/10.1016/j.jnt.2014.11.004.

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14

Dittmer, Samuel J. "Spoof odd perfect numbers". Mathematics of Computation 83, nr 289 (25.10.2013): 2575–82. http://dx.doi.org/10.1090/s0025-5718-2013-02793-7.

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15

KNOPFMACHER, ARNOLD, i FLORIAN LUCA. "ON PRIME-PERFECT NUMBERS". International Journal of Number Theory 07, nr 07 (listopad 2011): 1705–16. http://dx.doi.org/10.1142/s1793042111004447.

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We prove that the Diophantine equation [Formula: see text] has only finitely many positive integer solutions k, p1, …, pk, r1, …, rk, where p1, …, pk are distinct primes. If a positive integer n has prime factorization [Formula: see text], then [Formula: see text] represents the number of ordered factorizations of n into prime parts. Hence, solutions to the above Diophantine equation are designated as prime-perfect numbers.

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16

CHEN, SHI-CHAO, i HAO LUO. "ODD MULTIPERFECT NUMBERS". Bulletin of the Australian Mathematical Society 88, nr 1 (6.11.2012): 56–63. http://dx.doi.org/10.1017/s0004972712000858.

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AbstractA natural number $n$ is called multiperfect or $k$-perfect for integer $k\ge 2$ if $\sigma (n)=kn$, where $\sigma (n)$ is the sum of the positive divisors of $n$. In this paper, we establish a theorem on odd multiperfect numbers analogous to Euler’s theorem on odd perfect numbers. We describe the divisibility of the Euler part of odd multiperfect numbers and characterise the forms of odd perfect numbers $n=\pi ^\alpha M^2$ such that $\pi \equiv \alpha ~({\rm mod}~8)$, where $\pi ^\alpha $ is the Euler factor of $n$. We also present some examples to show the nonexistence of odd perfect numbers of certain forms.

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17

Nelsen, Roger B. "Proof Without Words: Perfect Numbers and Triangular Numbers". College Mathematics Journal 47, nr 3 (maj 2016): 171. http://dx.doi.org/10.4169/college.math.j.47.3.171.

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18

Ibro, Vait, i Eugen Ljajko. "Prime, perfect and friendly numbers". Zbornik radova Uciteljskog fakulteta Prizren-Leposavic, nr 12 (2018): 29–39. http://dx.doi.org/10.5937/zrufpl1812029i.

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19

Das, Bhabesh, i Helen K. Saikia. "On Near 3−Perfect Numbers". Sohag Journal of Mathematics 4, nr 1 (1.01.2017): 1–5. http://dx.doi.org/10.18576/sjm/040101.

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20

Beldon, Tom, i Tony Gardiner. "Triangular Numbers and Perfect Squares". Mathematical Gazette 86, nr 507 (listopad 2002): 423. http://dx.doi.org/10.2307/3621134.

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21

Sándor, József. "On certain rational perfect numbers". Notes on Number Theory and Discrete Mathematics 28, nr 2 (12.05.2022): 281–85. http://dx.doi.org/10.7546/nntdm.2022.28.2.281-285.

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22

Flowers, Joe. "Some Characterizations of Perfect Numbers". Missouri Journal of Mathematical Sciences 7, nr 3 (październik 1995): 104–15. http://dx.doi.org/10.35834/1995/0703104.

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23

Popov, Michael A. "On Plato's periodic perfect numbers". Bulletin des Sciences Mathématiques 123, nr 1 (styczeń 1999): 29–31. http://dx.doi.org/10.1016/s0007-4497(99)80011-6.

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24

Kurokawa, Nobushige, i Wakayama Masato. "Zeta functions ofq-perfect numbers". Rendiconti del Circolo Matematico di Palermo 53, nr 3 (październik 2004): 381–89. http://dx.doi.org/10.1007/bf02875730.

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25

Bezuszka, Stanley J., i Margaret J. Kenney. "Even Perfect Numbers: (Update)2". Mathematics Teacher 90, nr 8 (listopad 1997): 628–33. http://dx.doi.org/10.5951/mt.90.8.0628.

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Perfect Numbers may appear perfectly useless; however, they do have redeeming features. Specifically, the pursuit of perfect numbers leads us to examine the history of mathematics very closely to locate information about the progression of mathematicians who have discovered and worked with them. Students assigned to produce a report on perfect numbers and their properties will uncover some fascinating episodes. Further, perfect numbers are part of the frontier of the technological age. They are woven into the mystique of the supercomputer. Anyone with an interest in computing can try to determine how the computer helps in the search for perfect numbers and what algorithms are used for this purpose.

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26

De Medts, Tom, i Attila Maróti. "Perfect numbers and finite groups". Rendiconti del Seminario Matematico della Università di Padova 129 (2013): 17–33. http://dx.doi.org/10.4171/rsmup/129-2.

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27

Dunham, William. "Odd Perfect Numbers: A Triptych". Mathematical Intelligencer 42, nr 1 (22.08.2019): 42–46. http://dx.doi.org/10.1007/s00283-019-09915-6.

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28

Luca, Florian. "Perfect fibonacci and lucas numbers". Rendiconti del Circolo Matematico di Palermo 49, nr 2 (maj 2000): 313–18. http://dx.doi.org/10.1007/bf02904236.

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29

Bhutani, Kiran R., i Alexander B. Levin. "Graceful numbers". International Journal of Mathematics and Mathematical Sciences 29, nr 8 (2002): 495–99. http://dx.doi.org/10.1155/s0161171202007615.

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We construct a labeled graphD(n)that reflects the structure of divisors of a given natural numbern. We define the concept of graceful numbers in terms of this associated graph and find the general form of such a number. As a consequence, we determine which graceful numbers are perfect.

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30

Gandhi, K. Raja Rama. "Note on Perfect Numbers and their Existence". Bulletin of Mathematical Sciences and Applications 3 (luty 2013): 15–19. http://dx.doi.org/10.18052/www.scipress.com/bmsa.3.15.

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This paper will address the interesting results on perfect numbers. As we know that,perfect number ends with 6 or 8 and perfect numbers had some special relation with primes. Hereone can understand that the reasons of relation with primes and existence of odd perfect numbers. If exists, the structures of odd perfect numbers in modulo.

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31

YUAN, PINGZHI. "AN UPPER BOUND FOR THE NUMBER OF ODD MULTIPERFECT NUMBERS". Bulletin of the Australian Mathematical Society 89, nr 1 (28.01.2013): 1–4. http://dx.doi.org/10.1017/s000497271200113x.

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AbstractA natural number $n$ is called $k$-perfect if $\sigma (n)= kn$. In this paper, we show that for any integers $r\geq 2$ and $k\geq 2$, the number of odd $k$-perfect numbers $n$ with $\omega (n)\leq r$ is bounded by $\left({\lfloor {4}^{r} { \mathop{ \log } \nolimits }_{3} 2\rfloor + r\atop r} \right){ \mathop{ \sum } \nolimits }_{i= 1}^{r} \left({\lfloor kr/ 2\rfloor \atop i} \right)$, which is less than ${4}^{{r}^{2} } $ when $r$ is large enough.

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32

Bravo, Jhon J., i Florian Luca. "Perfect Pell and Pell–Lucas numbers". Studia Scientiarum Mathematicarum Hungarica 56, nr 4 (grudzień 2019): 381–87. http://dx.doi.org/10.1556/012.2019.56.4.1440.

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Abstract The Pell sequence is given by the recurrence Pn = 2Pn−1 + Pn−2 with initial condition P0 = 0, P1 = 1 and its associated Pell-Lucas sequence is given by the same recurrence relation but with initial condition Q0 = 2, Q1 = 2. Here we show that 6 is the only perfect number appearing in these sequences. This paper continues a previous work that searched for perfect numbers in the Fibonacci and Lucas sequences.

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33

Sándor, József. "On certain rational perfect numbers, II". Notes on Number Theory and Discrete Mathematics 28, nr 3 (10.08.2022): 525–32. http://dx.doi.org/10.7546/nntdm.2022.28.3.525-532.

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We continue the study from [1], by studying equations of type $\psi(n) = \dfrac{k+1}{k} \cdot \ n+a,$ $a\in \{0, 1, 2, 3\},$ and $\varphi(n) = \dfrac{k-1}{k} \cdot \ n-a,$ $a\in \{0, 1, 2, 3\}$ for $k > 1,$ where $\psi(n)$ and $\varphi(n)$ denote the Dedekind, respectively Euler's, arithmetical functions.

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34

Kustin, Andrew R. "Perfect modules with Betti numbers (2,6,5,1)". Journal of Algebra 600 (czerwiec 2022): 71–124. http://dx.doi.org/10.1016/j.jalgebra.2022.02.005.

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35

CHEN, SHI-CHAO, i HAO LUO. "BOUNDS FOR ODD k-PERFECT NUMBERS". Bulletin of the Australian Mathematical Society 84, nr 3 (21.07.2011): 475–80. http://dx.doi.org/10.1017/s0004972711002462.

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AbstractLet k≥2 be an integer. A natural number n is called k-perfect if σ(n)=kn. For any integer r≥1, we prove that the number of odd k-perfect numbers with at most r distinct prime factors is bounded by (k−1)4r3.

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36

Becher, Verónica, i Olivier Carton. "Normal numbers and nested perfect necklaces". Journal of Complexity 54 (październik 2019): 101403. http://dx.doi.org/10.1016/j.jco.2019.03.003.

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37

Fletcher, S. Adam, Pace P. Nielsen i Pascal Ochem. "Sieve methods for odd perfect numbers". Mathematics of Computation 81, nr 279 (1.09.2012): 1753–76. http://dx.doi.org/10.1090/s0025-5718-2011-02576-7.

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38

Gallardo, Luis H., i Olivier Rahavandrainy. "New Congruences for Odd Perfect Numbers". Rocky Mountain Journal of Mathematics 36, nr 1 (luty 2006): 225–35. http://dx.doi.org/10.1216/rmjm/1181069496.

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39

Cai, Tianxin, Deyi Chen i Yong Zhang. "Perfect numbers and Fibonacci primes (I)". International Journal of Number Theory 11, nr 01 (24.11.2014): 159–69. http://dx.doi.org/10.1142/s1793042115500098.

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In this paper, we introduce the concept of F-perfect number, which is a positive integer n such that ∑d|n,d<n d2 = 3n. We prove that all the F-perfect numbers are of the form n = F2k-1 F2k+1, where both F2k-1 and F2k+1 are Fibonacci primes. Moreover, we obtain other interesting results and raise a new conjecture on perfect numbers.

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40

Lorenzini, Anna. "Betti numbers of perfect homogeneous ideals". Journal of Pure and Applied Algebra 60, nr 3 (październik 1989): 273–88. http://dx.doi.org/10.1016/0022-4049(89)90088-1.

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41

REN, XIAO-ZHI, i YONG-GAO CHEN. "ON NEAR-PERFECT NUMBERS WITH TWO DISTINCT PRIME FACTORS". Bulletin of the Australian Mathematical Society 88, nr 3 (11.03.2013): 520–24. http://dx.doi.org/10.1017/s0004972713000178.

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AbstractRecently, Pollack and Shevelev [‘On perfect and near-perfect numbers’, J. Number Theory 132 (2012), 3037–3046] introduced the concept of near-perfect numbers. A positive integer $n$ is called near-perfect if it is the sum of all but one of its proper divisors. In this paper, we determine all near-perfect numbers with two distinct prime factors.

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42

Goto, Takeshi. "Upper Bounds for Unitary Perfect Numbers and Unitary Harmonic Numbers". Rocky Mountain Journal of Mathematics 37, nr 5 (październik 2007): 1557–76. http://dx.doi.org/10.1216/rmjm/1194275935.

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43

Jakimczuk, Rafael. "Divisors of numbers with k prime factors and perfect numbers". International Mathematical Forum 10 (2015): 339–47. http://dx.doi.org/10.12988/imf.2015.5435.

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44

Asadulla, Syed. "Thirty-nine perfect numbers and their divisors". International Journal of Mathematics and Mathematical Sciences 9, nr 1 (1986): 205–6. http://dx.doi.org/10.1155/s016117128600025x.

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The following results concerning even perfect numbers and their divisors are proved: (1) A positive integernof the form2p−1(2p−1), where2p−1is prime, is a perfect number; (2) every even perfect number is a triangular number; (3)τ(n)=2p, whereτ(n)is the number of positive divisors ofn; (4) the product of the positive divisors ofnisnp; and (5) the sum of the reciprocals of the positive divisors ofnis2. Values ofpfor which 30 even perfect numbers have been found so far are also given.

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45

Yamada, Tomohiro. "Odd perfect numbers of a special form". Colloquium Mathematicum 103, nr 2 (2005): 303–7. http://dx.doi.org/10.4064/cm103-2-13.

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46

MAEDA, Takao, i Takafumi HAYASHI. "Parameterization of Perfect Sequences of Real Numbers". IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E94-A, nr 6 (2011): 1401–7. http://dx.doi.org/10.1587/transfun.e94.a.1401.

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47

Starni, Paolo. "On Dris conjecture about odd perfect numbers". Notes on Number Theory and Discrete Mathematics 24, nr 1 (marzec 2018): 5–9. http://dx.doi.org/10.7546/nntdm.2018.24.1.5-9.

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48

Dris, Jose Arnaldo Bebita, i Immanuel Tobias San Diego. "Some modular considerations regarding odd perfect numbers". Notes on Number Theory and Discrete Mathematics 26, nr 2 (lipiec 2020): 27–33. http://dx.doi.org/10.7546/nntdm.2020.26.2.27-33.

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49

MAEDA, Takao, i Takafumi HAYASHI. "Parameterization of Perfect Arrays of Real Numbers". IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E94-A, nr 11 (2011): 2178–87. http://dx.doi.org/10.1587/transfun.e94.a.2178.

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50

Ghusayni *, B. "Maple explorations, perfect numbers, and Mersenne primes". International Journal of Mathematical Education in Science and Technology 36, nr 6 (15.09.2005): 643–54. http://dx.doi.org/10.1080/00207390500064080.

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