Monday, December 30, 2013

Polynesian people used binary numbers 600 years ago (NATURE)

Nature | News
Polynesian people used binary numbers 600 years ago 
Base-2 system helped to simplify calculations centuries before Europeans rediscovered it.
Philip Ball - 16 December 2013

Sunday, December 29, 2013

The Trouble With Online College

The Trouble With Online College (NY Times, February 19, 2013 - page A22)
Education via the Internet has been overrated and could produce more dropouts than degrees

Friday, December 27, 2013

Eu - o genealogie matematica

O arheologie genealogica matematica a subsemnatului, pornind de la The Mathematics Genealogy Project - Mihai Caragiu:
Gregory Palamas
Nilos Kabasilas 1363
Demetrios Kydones
Georgios Plethon Gemistos 1380
Basilios Bessarion 1436
Johannes Argyropoulos 1444
Marsilio Ficino 1462
Angelo Poliziano 1477
Scipione Fortiguerra 1493
Girolamo (Hieronymus Aleander) Aleandro 1508
Rutger Rescius 1513
Johannes Winter von Andernach 1527
Johann Sturm
Pierre de La Ramée 1536
Theodor Zwinger 1553
Petrus Ryff 1584
Emmanuel Stupanus 1613
Franciscus de le Boë Sylvius 1637
Rudolf Wilhelm Krause 1671
Simon Paul Hilscher 1704
Johann Andreas Segner 1734
Johann Georg Büsch 1752
Johann Elert Bode
Johann Friedrich Pfaff 1786
Carl Friedrich Gauss 1799
Friedrich Wilhelm Bessel 1810
Heinrich Ferdinand Scherk 1823
Ernst Eduard Kummer 1831
Nicolai Vasilievich Bugaev 1866
Dimitri Fedorowitsch Egorov 1901
Nikolai Nikolayevich Luzin 1915
Aleksandr Yakovlevich Khinchin
Aleksandr Adolfovich Buchstab 1939
Ilya Piatetski-Shapiro 1954
Leonid Vaserstein 1969
Mihai Caragiu 1996

Monday, December 23, 2013

Dorneni de top: Cristian Cobeli

Cristi străbate oraşul pe jos sau cu bicicleta iar de fiecară dată îi surprind un surîs în colţul gurii. Un surîs care nu vine nici din superioritatea cercetătorului în ale matematicii şi nici din partea doctorului în matematică care a predat ani de zile la New York. Este surîsul unui om care a înţeles această lume, cu bunele şi relele ei, un om care-şi manifestă ortodoxismul în modul cel mai onest, cu palmele dacă este cazul, cum a fost la construcţia catedralei, sau inspirîndu-şi copiii la Clubul Copiilor, iniţiindu-i în jocul de Go. (Dorneni de top: Cristian Cobeli - Monitorul de Dorna - Mihai CIOATĂ, 8 Mai 2009)

Monday, December 2, 2013

An analogue of the Proth-Gilbreath conjecture (new paper)

O fenomenologie a numerelor prime - in gen fantasy cu demonstratii plus "experiment" (computer).

Far East Journal of Mathematical Sciences (FJMS)
Volume 81, Issue 1, Pages 1 - 12 (October 2013)
AN ANALOGUE OF THE PROTH-GILBREATH CONJECTURE 
Mihai Caragiu, Alexandru Zaharescu and Mohammad Zaki
Communicated by Juliusz Brzezinski


Saturday, November 30, 2013

Small pairs of primitive roots with sum and difference one - raw data for primes up to 1013



Prime p
Primitive roots
Category 1
Sum 1
Small (least < \/p )
Category 2
Difference 1
Small (least < \/p )
2
1
no
no
3
2
no
no
5
2, 3
no
(2,3)
7
3, 5
no
no
11
2, 6, 7, 8
no
no
13
2, 6, 7, 11
no
no
17
3, 5, 6, 7, 10, 11, 12, 14
no
no
19
2, 3, 10, 13, 14, 15
no
(2,3)
23
5, 7, 10, 11, 14, 15, 17, 19, 20, 21
no
no
29
2, 3, 8, 10, 11, 14, 15, 18, 19, 21, 26, 27
(3,27)
(2,3)
31
3, 11, 12, 13, 17, 21, 22, 24
no
no
37
2, 5, 13, 15, 17, 18, 19, 20, 22, 24, 32, 35
no
no
41
6, 7, 11, 12, 13, 15, 17, 19, 22, 24, 26, 28, 29, 30, 34, 35
no
no
43
3, 5, 12, 18, 19, 20, 26, 28, 29, 30, 33, 34
no
no
47
5, 10, 11, 13, 15, 19, 20, 22, 23, 26, 29, 30, 31, 33, 35, 38, 39, 40, 41, 43, 44, 45
(5,43)
no
53
2, 3, 5, 8, 12, 14, 18, 19, 20, 21, 22, 26, 27, 31, 32, 33, 34, 35, 39, 41, 45, 48, 50, 51
(3,51)
(2,3)


From now on, the set of all primitive roots (column 2 above) is not listed.



59
(6, 54)
no
61
(7, 55)
(6, 7)
67
(7, 61)
no
71
(7, 65)
no
73
no
no
79
(3, 77)
(6, 7)
83
(5, 79)
(5, 6)
89
(7, 83)
(6, 7)
97
no
no
101
(3, 99)
(2, 3)
103
(5, 99)
(5, 6)
107
(5, 103)
(5, 6)
109
no
(10, 11)
113
(6, 108)
(5, 6)
127
no
(6, 7)
131
(6, 126)
no
137
(6, 132)
(5, 6)
139
no
(2, 3)
149
(3, 147)
(2, 3)
151
(6, 146)
(6, 7)
157
(6, 152)
(5, 6)
163
(11, 153)
(11, 12)
167
(5, 163)
no
173
(3, 171)
(2, 3)
179
(6, 174)
(6, 7)
181
no
no
191
no
no
193
no
no
197
(3, 195)
(2, 3)
199
(3, 197)
no
211
(7, 205)
(2, 3)
223
(10, 214)
(5, 6)
227
(5, 223)
(13, 14)
229
(7, 223)
(6, 7)
233
(6, 228)
(5, 6)


Golomb construction


For the prime 47, and primitive roots 5 and 43 with sum 1 mod 47, the Golomb-type Costas array is defined as follows: a(I,J)=1 if 5^I+43^J=1 (mod 47), otherwise a(I,J)=0. 
MATLAB style (row #1 @ the bottom) - granted, an imperfect screen shot...
The displacement vectors between pairs of distinct points are all distinct, which makes it useful, for example, in radar design.