Chuyện Đời , chuyện Người và nghiệp Toán: tháng 4 2017

What is a Mersenne Prime Number?

A Mersenne Number is a number that is equal to one less than a power of 2, or (2^n-1) where n is any positive integer. Mersenne numbers are named after Marin Mersenne, a French monk.

A Mersenne Prime Number is a prime number that happens to also be a Mersenne Number. A Mersenne Prime Number will always have a value equal to (2^n-1) where n is one of a selected list of positive prime numbers.

When I discovered that there was a formula for "Mersenne" prime numbers, I wrote another program for that, which gave me these results and is listed later on this page:

2^2-1 = 3
2^3-1 = 7
2^5-1 = 31
2^7-1 = 127
2^13-1 = 8191
2^17-1 = 131,071
2^19-1 = 524,287
2^31-1 = 2,147,483,647

As mentioned above, every known "perfect number" can be generated from a "Mersenne prime". If (2^n-1) describes a "Mersenne prime," then the matching "perfect number" is equal to:

2 ^ (n-1) * ( 2^n - 1)

where n = 2, 3, 5, 7, 13, 17, 19, 31 or other "mersenne" exponents. All of the known "Mersenne" exponents are listed in one of the sections above.

Can we prove that every "Mersenne Prime Number" can generate a corresponding Perfect number? Yes, we can.

Can we prove that every possible "perfect" number can be generated from a "Mersenne" prime? No, we cannot. (I'm hoping that someday someone will find an ODD "perfect" number, and an odd "perfect" number can NOT come from a "Mersenne" prime.)

List of Known Mersenne Prime Numbers

#	2^p-1	Digits	Date Discovered	Discovered By	Method / Hardware	Perfect Number
1	2²-1	1	c. 500 BCE	Ancient Greek mathematicians		2¹ · (2²-1)
2	2³-1	1	c. 500 BCE	Ancient Greek mathematicians		2² · (2³-1)
3	2⁵-1	2	c. 275 BCE	Ancient Greek mathematicians		2⁴ · (2⁵-1)
4	2⁷-1	3	c. 275 BCE	Ancient Greek mathematicians		2⁶ · (2⁷-1)
5	2¹³-1	4	1456	Anonymous	trial division	2¹² · (2¹³-1)
6	2¹⁷-1	6	1588	Pietro Cataldi	trial division	2¹⁶ · (2¹⁷-1)
7	2¹⁹-1	6	1588	Pietro Cataldi	trial division	2¹⁸ · (2¹⁹-1)
8	2³¹-1	10	1772	Leonhard Euler	Enhanced trial division	2³⁰ · (2³¹-1)
9	2⁶¹-1	19	1883	Ivan Mikheevich Pervushin	Lucas sequences	2⁶⁰ · (2⁶¹-1)
10	2⁸⁹-1	27	1911 Jun	R. E. Powers	Lucas sequences	2⁸⁸ · (2⁸⁹-1)
11	2¹⁰⁷-1	33	1914 Jun 11	R. E. Powers	Lucas sequences	2¹⁰⁶ · (2¹⁰⁷-1)
12	2¹²⁷-1	39	1876 Jan 10	Édouard Lucas	Lucas sequences	2¹²⁶ · (2¹²⁷-1)
13	2⁵²¹-1	157	1952 Jan 30	Raphael M. Robinson	L-L / SWAC	2⁵²⁰ · (2⁵²¹-1)
14	2⁶⁰⁷-1	183	1952 Jan 30	Raphael M. Robinson	L-L / SWAC	2⁶⁰⁶ · (2⁶⁰⁷-1)
15	2^1,279-1	386	1952 Jun 25	Raphael M. Robinson	L-L / SWAC	2^1,278 · (2^1,279-1)
16	2^2,203-1	664	1952 Oct 07	Raphael M. Robinson	L-L / SWAC	2^2,202 · (2^2,203-1)
17	2^2,281-1	687	1952 Oct 09	Raphael M. Robinson	L-L / SWAC	2^2,280 · (2^2,281-1)
18	2^3,217-1	969	1957 Sep 08	Hans Riesel	L-L / BESK	2^3,216 · (2^3,217-1)
19	2^4,253-1	1,281	1961 Nov 03	Alexander Hurwitz	L-L / IBM 7090	2^4,252 · (2^4,253-1)
20	2^4,423-1	1,332	1961 Nov 03	Alexander Hurwitz	L-L / IBM 7090	2^4,422 · (2^4,423-1)
21	2^9,689-1	2,917	1963 May 11	Donald B. Gillies	L-L / ILLIAC II	2^9,688 · (2^9,689-1)
22	2^9,941-1	2,993	1963 May 16	Donald B. Gillies	L-L / ILLIAC II	2^9,940 · (2^9,941-1)
23	2^11,213-1	3,376	1963 Jun 02	Donald B. Gillies	L-L / ILLIAC II	2^11,212 · (2^11,213-1)
24	2^19,937-1	6,002	1971 Mar 04	Bryant Tuckerman	L-L / IBM 360/91	2^19,936 · (2^19,937-1)
25	2^21,701-1	6,533	1978 Oct 30	Landon Curt Noll & Laura Nickel	L-L / CDC Cyber 174	2^21,700 · (2^21,701-1)
26	2^23,209-1	6,987	1979 Feb 09	Landon Curt Noll	L-L / CDC Cyber 174	2^23,208 · (2^23,209-1)
27	2^44,497-1	13,395	1979 Apr 08	Harry Lewis Nelson & David Slowinski	L-L / Cray 1	2^44,496 · (2^44,497-1)
28	2^86,243-1	25,962	1982 Sep 25	David Slowinski	L-L / Cray 1	2^86,242 · (2^86,243-1)
29	2^110,503-1	33,265	1988 Jan 28	Walter Colquitt & Luke Welsh	L-L / NEC SX-2	2^110,502 · (2^110,503-1)
30	2^132,049-1	39,751	1983 Sep 19	David Slowinski	L-L / Cray X-MP	2^132,048 · (2^132,049-1)
31	2^216,091-1	65,050	1985 Sep 01	David Slowinski	L-L / Cray X-MP/24	2^216,090 · (2^216,091-1)
32	2^756,839-1	227,832	1992 Feb 19	David Slowinski & Paul Gage	L-L / Maple on Harwell Lab Cray-2	2^756,838 · (2^756,839-1)
33	2^859,433-1	258,716	1994 Jan 04	David Slowinski & Paul Gage	L-L / Cray C90	2^859,432 · (2^859,433-1)
34	2^1,257,787-1	378,632	1996 Sep 03	David Slowinski & Paul Gage	L-L / Cray T94	2^1,257,786 · (2^1,257,787-1)
35	2^1,398,269-1	420,921	1996 Nov 13	GIMPS / Joel Armengaud	L-L / Prime95 on 90 MHz Pentium PC	2^1,398,268 · (2^1,398,269-1)
36	2^2,976,221-1	895,932	1997 Aug 24	GIMPS / Gordon Spence	L-L / Prime95 on 100 MHz Pentium PC	2^2,976,220 · (2^2,976,221-1)
37	2^3,021,377-1	909,526	1998 Jan 27	GIMPS / Roland Clarkson	L-L / Prime95 on 200 MHz Pentium PC	2^3,021,376 · (2^3,021,377-1)
38	2^6,972,593-1	2,098,960	1999 Jun 01	GIMPS / Nayan Hajratwala	L-L / Prime95 on 350 MHz Pentium II IBM Aptiva	2^6,972,592 · (2^6,972,593-1)
39	2^13,466,917-1	4,053,946	2001 Nov 14	GIMPS / Michael Cameron	L-L / Prime95 on 800 MHz Athlon Thunderbird	2^13,466,916 · (2^13,466,917-1)
40	2^20,996,011-1	6,320,430	2003 Nov 17	GIMPS / Michael Shafer	L-L / Prime95 on 2 GHz Dell Dimension	2^20,996,010 · (2^20,996,011-1)
41	2^24,036,583-1	7,235,733	2004 May 15	GIMPS / Josh Findley	L-L / Prime95 on 2.4 GHz Pentium 4 PC	2^24,036,582 · (2^24,036,583-1)
42	2^25,964,951-1	7,816,230	2005 Feb 18	GIMPS / Martin Nowak	L-L / Prime95 on 2.4 GHz Pentium 4 PC	2^25,964,950 · (2^25,964,951-1)
43	2^30,402,457-1	9,152,052	2005 Dec 15	GIMPS / Curtis Cooper & Steven Boone	L-L / Prime95 on 2 GHz Pentium 4 PC	2^30,402,456 · (2^30,402,457-1)
44	2^32,582,657-1	9,808,358	2006 Sep 04	GIMPS / Curtis Cooper & Steven Boone	L-L / Prime95 on 3 GHz Pentium 4 PC	2^32,582,656 · (2^32,582,657-1)
45	2^37,156,667-1	11,185,272	2008 Sep 06	GIMPS / Hans-Michael Elvenich	L-L / Prime95 on 2.83 GHz Core 2 Duo PC	2^37,156,666 · (2^37,156,667-1)
46*	2^42,643,801-1	12,837,064	2009 Jun 04	GIMPS / Odd M. Strindmo	L-L / Prime95 on 3 GHz Core 2 PC	2^42,643,800 · (2^42,643,801-1)
47*	2^43,112,609-1	12,978,189	2008 Aug 23	GIMPS / Edson Smith	L-L / Prime95 on Dell Optiplex 745	2^43,112,608 · (2^43,112,609-1)
48*	2^57,885,161-1	17,425,170	2013 Jan 25	GIMPS / Curtis Cooper	L-L / Prime95 on Intel Core2 Duo E8400 @ 3.00GHz	2^57,885,160 · (2^57,885,161-1)
49*	2^74,207,281-1	22,338,618	2016 Jan 07	GIMPS / Curtis Cooper	L-L / Prime95 on Intel i7-4790 @ 3.60GHz	2^74,207,280 · (2^74,207,281-1)

Chuyện Đời , chuyện Người và nghiệp Toán

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