นอกจากจะเป็นนักคณิตศาสตร์ที่โดดเด่น เขายังเป็นชายผู้มีเอกลักษณ์ ในเรื่องของการสร้างคำศัพท์แบบที่เป็นเอกลักษณ์อย่างยิ่ง ลองดูตัวอย่างของสิ่งที่เขา “นิยาม” ขึ้นมาบ้างดีกว่า

- เด็กๆ ถูกเรียกว่า epsilons(ε) เพราะว่าในทางคณิตศาสตร์โดยเฉพาะในแคลคูลัส มันคือจำนวนที่เป็นบวกซึ่งมีขนาดเล็กมากๆ
- ผู้หญิง คือ “เจ้านาย”แต่ ผู้ชาย คือ “ทาส”
- คนที่ “ตายแล้ว” คือคนที่หยุดการทำคณิตศาสตร์ ส่วนคนที่ “หมดลมหายใจ” คือคนที่ “จากไป”
- เครื่องดื่มแอลกอฮอล์สำหรับเขา คือ “ยาพิษ” และดนตรีคือ “เสียงหนวกหู”
- คนที่แต่งงานกัน จะถูกเรียกว่า “โดนจับ” แต่คนที่หย่าร้าง จะถูกเรียกว่า “มีอิสรภาพ”
- การสอนคณิตศาสตร์ คือ “การเทศนา” และการให้ทำข้อสอบปากเปล่าแก่นักศึกษา คือ “การทรมาน”

Paul Erdős (occasionally spelled Erdos or Erdös; Hungarian: Erdős Pál; 26 March 1913 – 20 September 1996) was an immensely prolific and famously eccentric Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory.

Paul Erdős(เกิดในวันที่ 26 ของเดือนมีนาคม ปี 1913 และเสียชีวิตในวันที่ 20 กันยายน 1996) คือนักคณิตศาสตร์ชาวฮังกาเรียนผู้เป็นที่รู้จักอย่างกว้างขวางและได้รับการนับถือในวงการคณิตศาสตร์อย่างยิ่ง เขาได้ตีพิมพ์งานวิจัยทางคณิตศาสตร์มากกว่านักคณิตศาสตร์คนใดในประวัติศาสตร์ยกเว้นเพียง ออยเลอร์เท่านั้น และทำงานร่วมกับเพื่อนร่วมงานนับร้อยคน สิ่งซึ่งเป็นที่สนใจพิเศษของเขา คือ คอมบินาทอริก, ทฤษฎีกราฟ, ทฤษฎีจำนวน,  คณิตศาสตร์วิเคราะห์, ทฤษฎีการประมาณค่า, ทฤษฎีของเซต และ ทฤษฎีของความน่าจะเป็น

Nokia 8800 series of luxury phones have always focused on high quality materials and superior build quality, while holding back on the spec sheet. With the Nokia 8800 Erdos this reputation is about to change.

First off, “Erdos” is just a code name so expect that to change at some point. And second – this won’t be an Arte. The Arte phones have had largely the same design, differing in materials only. The 8800 Erdos changes not only the design but also the way the phones are built.

The Nokia Ardos body will be constructed out of a single piece of “bullet silver”. This apparently is marketing talk for stainless steel, so you probably can’t kill a werewolf with it, but carving out the body out of a single piece of metal is all the rage with high-end laptops and the ridiculously expensive luxury phones.

The screen of the Erdos however is covered with regular scratch-proof glass, rather than sapphire, which is the usual weapon of choice in such cases. The glass covering the screen though turns silver when the display is off, giving the impression that the phone is a single slab of stainless steel. The screen itself is an OLED unit with QVGA resolution and a diagonal of 2.4 inches.

Nokia 8800 Erdos

The high-end design of the Nokia 8800 Erdos includes a rather unique feature too – it’s a slider like the rest of its siblings, but the keypad raises to the level of the display when you slide it out. This should make the keys of the top row much more accessible.

Sounds good already, but we haven’t even gotten to the good part yet – the Nokia 8800 Erdos is the first of its breed to be a full-on smartphone. It’ll run the Symbian OS S60 and feature Wi-Fi and A-GPS. The internal memory is 8GB, though there’s no card slot.

The camera is the best in the luxury class with its 5 megapixels and Carl Zeiss lens with autofocus. There’s a dual-LED flash too.

Rounding off the specs is the reasonably compact size of the phone. The 8800 Erdos measures 116.9 x 50,3 x 15.0 mm, though its weight is kept on the hush-hush. There is also a dedicated noise reduction microphone on the back.

The Nokia 8800 Erdos will be available in October with tri-band GSM (900/1800/1900) and dual-band 3G/HSPA. The price is the other thing missing from the rumored specs, though this could easily top the price point of the Gold Arte.

Finlandezii de la Nokia pregatesc un nou telefon din seria 8800: Erdos. Acesta va dispune de un ecran OLED de 2.4 inch, WiFi, 3G, GPS si multe altele. Nu se stie cu exactitate data lansarii. Pentru moment putem privi insa filmul de prezentare.

Alttaki videoda da görebileceğiniz Erdos‘un fanatik bir tasarım maketinden öte bir ürün olduğu ile ilgili hiçbir resmi veri yok, fakat inanmak bize bir şey kaybettirmez, değil mi? Dürüst olmak gerekirse piyasaya sürülmesi  gerçekten söylendiği kadar yakınsa Nokia’nın bu “mücevheri” yakın zamanda yapılmış olan Nokia World fuarında göstermemeyi seçme sebebini tahmin etmek çok zor. Yine de, elimize ulaşan bilgi kırıntıları içinde OLED’li Erdos‘un gelecek ay gibi yakın bir zamanda dünya yüzü göreceği de var.

8xxx serisinden (8800 gibi) bir parçaymış gibi duran bu ultra ince ve ultra parlak telefon paslanmaz çelikten. A-GPS, 3G desteği, 2.4 inç QVGA (320 x 240) ekran, Wi-Fi, USB, 5 MP kamera (video kayıt desteği ile) ve 8Gb hafıza ile gelen Erdos eğer sadece bir fan hayalgücünün ürünü çıkarsa, yerine pekala X6′yı da koyabilirsiniz.

[Engadget]

Много информация заля интернет последните 24ч. относно предполагаемото ново устройство от финландците от Nokia. Колко всъщност от тази информация е истина – никой не знае. Със сигурност доста потребители , и може би Вие , които точно в този момент четете тази публикация си задавате подобен въпрос , дали наистина апаратът ще види бял свят ? Първоначално замислен като концепция от видето , което бе считано за просто фенщина (от фен) , с лъскавия си вид , OLED дисплей , 5mpx камера и т.н. няма да се впускам в детаили относно характеристиките на апарата , тъй като има предостатъчно информация , за който я потърси. За разлика от това , което всички (буквално ВСИЧКИ) други уеб сайтове са направили , а именно да покажат характеристиките на телефончето , да разкажат , че можело да бъде наследник на 8800 и т.н. всичката тази информация всъщност идваща от англоговорящи сайтове , където дори не се знае дали информацията е 100%-та истина , те просто превеждат и представят информацията на български език за тези , които не знаят английски език , че и дори не предоставят от къде са взели самата информация , което всъщност е в нарушение на доста лиензионни права , които повечето от тези уебсайтове притежават. Факт е , и че българинът не е свикнал да действа както е прието , не само по “европейски” , но и по “интернетски” , ако мога така да се изразя.

Аз ще дам просто моите 20 стотинки , или иначе казано ще кажа 2 думи по въпроса какво мисля Аз. Тъй като видях само копиране на информация , но и никъде не видях собствено мнение относно апарата , видеото и като цяло за информацията заляла мрежата. За финал ще добавя , така гледаните вече от всички потребители характеристики на парата + лъскавото концептуално видео.

Моите 20 стотинки: факт е , че телефонът наистина може да се появи на бял свят , факт е че , “българчето” обича лъскави дрешки , колички , телефончета , джунджурийки и т.н. , но защо ? Защо точно мобилните устройства Nokia , и специално серията 8ххх (да не е 8800 , тъй като цялата серия е 8ххх , за информация на някои не чак толкова информирани автори) , вдига толкова шум и врява сред интернет потребителите и то по-специално българските ? Отговорът е много простичък и се състои само в 5 буквички – N O K I A. Коментарите ще ги оставя на Вас (ако изобщо ви има) , тъй като не мисля да продължавам в тази насока.

И все пак – ако устройството види бял свят , то е предвидено да се състои много скоро – месец Октомври , което отново не е официална , 100%-та доказана реална информация , тъй като много хора не знаят , но наскоро се проведе т.н. Nokia World expo , на което този модел не бе представен. При всички положения , съвсем скоро ще разберем дали видеото ще стане реалност , дали поредните “луксманияци” ще му се радват , и дали ще удари космическите суми (разбира се , че да).

Техническите характеристики (видео след тях) според уебсайта cellpassion.com са както следва:

Dimensions: 116.9 x 50,3 x 15.0 mm
Display main: QVGA 320×240 pixels, 16M colours, OLED with 2,4″ active area
Protocols: WCDMA bands: I + VIII, GSM 900/1800/1900
Connectivity: WLAN, AGPS, High Speed USB, 30 Mbit/sec (micro-USB interface inclusive charging), Stereo Bluetooth ver 2.0 (A2DP & AVRCP)
Camera: 5 Mpix Carl Zeiss AF with dual LED
Video performance: 288×352 @ 30 fps // 480×640 @ 15 fps
Memory: 8 GB internal non user replacable

via: Google , cellpassion.com , Engadget

We are going to tell a strange yarn revolving around Erdos numbers  and an introverted Scandinavian mathematician: Atle Selberg.

When Nazi Germany occupied Norway, he remained isolated from the rest of the scientific community . In those years, he gained a demonstration of the Dirichlet’s Theorem  (without the RH) and he approached a demonstration of the theorem of prime numbers of Gauss.

In his solitude Selberg had never published with anyone, nor ever discussed his ideas with other mathematicians. But ones upon a time… he  became a friend of Paul Turan, and Atle revealed him some of their results. Turan, for misfortune of Selberg, was a great friend of Erdos, and inevitably the ideas of Atle reached the ears of the hungarian mathematician.

Once, Erdos’ friendship  saved the Turan’s lives. Indeed, in 1945, the mathematician was caught by Russian troops in Budapest, recently released, in possession of many sheets containing strange formulas and codes, that seemed Nazi coded instructions! Turan managed to survive only because the good Erdos testified for him.

Came back to us. Erdos,  barely aware of Atle’s results, used them to prove a generalized version of Bertrand’s postulate. Selberg was happy: it was the missing card to prove the theorem of prime numbers!

Selberg confided his result to Erdos, who immediately began to spread the notice in the world, pressing Atle for a four hands article. In a short time, thanks to an extensive network of correspondence and conferences, the news of the demonstration came in every faculties of mathematics in the world. Unfortunately, however, someone began to say that the theorem was proved by Erdos. Once, indeed, a mathematician  said to Selberg, ignoring who it was: “Have you heard? Erdos and a scandinavian mathematician prooved the theorem of prime numbers! “

It was the spark that made the vase overflow … Selberg locked himself again in the solitude, hating Erdos for the rest of the life.

Throughout Selberg has collaborated once with another mathematician, signing with him an article. He was Saradavam Chowla, who had Erdos number of 1.

Thus Selberg obtained EN 2.

(< Previous) (Next >)

Una delle vicende più strambe sui numeri di Erdos ruota attorno ad un introverso matematico scandinavo: Atle Selberg.

Questi rimase isolato dal resto della comunità scientifica quando la Germania nazista occupò la Norvegia. In questi anni migliorò una dimostrazione di Dirichlet (senza passare dall’Ipotesi di Riemann) e si avvicinò a una dimostrazione elementare del Teorema dei numeri primi di Gauss.

Nella sua solitudine Selberg non aveva mai pubblicato con nessuno, né mai discusso delle proprie idee con altri matematici all’altezza. Fu solo un caso che lo portò a diventare amico di Paul Turan e a rivelargli parte dei propri risultati. Turan, per sfortuna di Selberg, era un grande amico di Erdos, e inevitabilmente le idee di Atle giunsero alle orecchie del matematico ungherese.

Una volta, l’amicizia di Erdos salvò la vita a Turan. Infatti, nel 1945, il matematico fu “beccato” dalle truppe russe mentre vagava per Budapest, da poco liberata, in possesso di molti fogli contenenti strane formule e codici che parevano proprio istruzioni tedesche codificate! Turan riuscì a salvarsi solo perchè il buon Erdos testimoniò per lui.

Ma torniamo a noi. Appena Erdos venne a conoscenza dei risultati di Atle li utilizzò per dimostrare una versione generalizzata del postulato di Bertrand. Selberg ne fu felice: era la tessera che gli mancava per dimostrare il Teorema dei numeri primi!

Selberg confidò il proprio risultato a Erdos, il quale subito iniziò a spargere la voce e insistere per un articolo a quattro mani. Grazie a una fitta rete epistolare e a conferenze in breve tempo la notizia della dimostrazione entrò in tutte le facoltà di matematica del mondo. Purtroppo però si iniziava a dire in giro che il teorema era stato dimostrato da Erdos. Una volta, addirittura, un matematico disse a Selberg, non sapendo chi quest’ultimo fosse: “Hai sentito? Erdos e un matematico scandinavo hanno dimostrato il Teorema dei numeri primi!”

Fu la scintilla che fece traboccare il vaso… Selberg si rinchiuse nuovamente nella propria solitudine, odiando Erdos per il resto della vita.

In tutto Selberg ha collaborato una  sola volta con un altro matematico, firmando con lui un articolo. Questi era Saradavam Chowla, che aveva numero di Erdos 1.

Fu così che Selberg ottenne numero di Erdos 2.

(< Precedente) (Successivo >)

This is the first of a series of articles about Erdos’ numbers (EN).  We will understand what they are, then we’ll deal with not-mathematics versions, the concept of scale-free network, and some amusing anecdotes.

Paul Erdos was a Hungarian mathematician. A strange person, but a genius. He worked in many different fields: combinatorics, graph theory, number theory, analysis, numerical, …

Thanks to his way of doing and his eccentricity, Paul is certainly one of the most popular mathematics of the twentieth century (he died in the nineties). I talked with italian mathematicians who met him during some conferences in Italy and they have reported that it was a wonderful person. An applicant’s sentence has now become, in our environment, a real say, and even the summary of the mathematician lifestyle:

“A mathematician is a device for turning coffee into theorems.”

After Euler, Erdos is the mathematician who has produced more publications: it is estimated that 511 different mathematicians have had the honor to sign an article with him.

We are not going to explore his main results, concentrating ourselfes on the Erdos’ numbers. The sequence counts the publication distance from the great mathematician. Naturally Erdos is awarded of EN 0, while the 511 mathematicians who have published with him has EN 1. In the same way, mathematicians who have published with at least one of these 511 have EN 2, and so forth. Currently we know that more than 8000 people may have Erdos number of 2.

Of course there are mathematicians in the past (prior to Erdos) owning an EN, eg Frobenius (the morphisms one) EN is 3, Dedekind (the sections one) EN is 7, and it seems that Gauss and Euler (we don’t need to introduce them) have an EN too.

The baseball player Hank Aaron was one of the 511, having been the subject of a mathematical study and having signed with Erdos a baseball ball.

(Next >)

Inizia una serie di articoli-curiosità che hanno come pretesto i numeri di Erdos. In questo avrete modo di conoscerli, mentre nei successivi incontrerete varianti non matematiche, il concetto di rete ad invarianza di scala e alcuni divertenti aneddoti ad essi correlati.

Paul Erdos è stato un grande matematico ungherese. Un tipo un po’ strano, ma un genio. Ha lavorato in moltissimi campi diversi: combinatorica, teoria dei grafi, teoria dei numeri, analisi, numerica, …

Grazie al suo modo di fare e alla sua eccentricità, Paul è certamente uno dei matematici più amati del XX Secolo (è morto negli anni novanta). Era conosciuto ovunque perché vagava per il mondo facendosi ospitare in casa da colleghi con i quali produceva stupendi articoli. Conosco matematici che son stati a conferenze con lui in Italia e mi han riferito che era una persona stupenda. Una sua frase ricorrente è ormai diventata, nel nostro ambiente, un vero e proprio detto, se non addirittura il motto riassuntivo dello stile di vita matematico:

“I matematici sono macchine per trasformare caffè in teoremi.”

Dopo Euler, Erdos è il matematico che ha prodotto più pubblicazioni: si è calcolato che ben 511 matematici (e non solo) han avuto l’onore di firmare un articolo con lui.

Non ci dilunghiamo sulla portata delle sue scoperte (che comunque avrete modo di conoscere in altri articoli) e ci concentriamo sui numeri di Erdos. La successione composta da questi conta la distanza di pubblicazione dal grande matematico, ovvero è stato assegnato numero di Erdos 0 a Erdos, 1 ai 511 matematici che han pubblicato con lui, 2 ai matematici che han pubblicato con almeno uno dei suddetti 511 (senza far parte di questi), e così via. Attualmente sappiamo che più di 8000 persone possono vantare numero di Erdos 2.

Naturalmente ci sono matematici del passato (precedenti ad Erdos) dotati del numero di Erdos: ad esempio Frobenius (sì, quello del morfismo) ha il 3, Dedekind (quello delle sezioni) il 7, e pare che anche Gauss ed Eulero (qui non servono predentazioni…) ne posseggano uno.

Il giocatore di baseball Hank AAron è uno dei 511 a vantare numero di Erdos 1, essendo stato oggetto di uno studio del matematico ed avendo firmato con lui una pallina da baseball.

(Successivo >)

Apokalipsa/ źródło xkcd.com

Po opublikowaniu wczoraj tego komiksu, rechot matematyków słychać było na kampusie przez cały dzień. Oto historia liczb Erdösa:

Paul Erdös był wybitnym węgierskim matematykiem, prawdopodobnie jednym z najbardziej twórczych i wszechstronnych badaczy w  historii matematyki (a przynajmniej historii nowożytnej, dla której możemy użyć liczby i znaczenia publikacji jako miary – w szranki mógł iść jedynie z Eulerem). Wśród jego – bardzo bardzo licznych – osiągnięć znajduje się praca nad teorią Ramseya, praca nad zastosowaniem metody probabilistycznej. Wykazał się i wpisał do annałów historii dzięki swojej pracy w zakresie kombinatoryki, teorii grafów, teorii liczb, analizy matematycznej, teorii przybliżeń, teorii mnogości oraz teorii prawdopodobieństwa. Innymi słowy: jeśli jeszcze gdzieś w matematyce dało się dorzucić trzy grosze, Paul Erdös swoje dorzucił.

Nie wszyscy mu współcześni oczywiście go kochali. Erdös prezentował pewien specyficzny typ badacza – nie spędzał wielu lat na rozwijanie i rozbudowywanie jednej teorii, lecz koncentrował się na jednym, wybranym problemie w ramach jakiejś dziedziny – a jak widzieliśmy żadna gałąź matematyki mu się nie oparła – rozwiązywał go i ruszał dalej. Niemniej, i tu dochodzimy powoli do liczb Erdösa, nie był on egoistą. Zmagając się z jakimś problemem zawsze współpracował z innymi matematykami, z którymi oczywiście później współnie publikował. Ktoś może oczywiście powiedzieć, że był to rodzaj pasożytnictwa – wręcz jednak przeciwnie! Wielu z tych ludzi pewnie nigdy by nie zaistniało, gdyby nie współpraca z Erdösem, a matematycy po dziś dzień chwalą się, jeśli mieli okazję z nim współpracować. A należy podkreślić, że ten jego pęd do współpracy, to nie było zachowanie typowe, bo w matematyce, jak w żadnej innej dziedzinie, tej współpracy tak wiele nie ma (być może dlatego, że ta często wynika z kompromisów – ja mam taki instrument, ty masz inny, razem możemy zdziałać coś naprawdę wielkiego. W matematyce często wystarczy kawałek kredy i tablica).

Z tych jego jakże licznych współprac (a miał Erdös w całym swoim życiu 511 współpracowników!) zrodził się mit liczb Erdösa, ten najsłynniejszy bodaj matematyczny folklor. Liczba Erdösa określe jak blisko współpracy z Erdösem była dana osoba. Sam Erdös ma więc liczbę Erdösa równą zeru. Osoby, które były współautorami publikacji wraz z Węgrem, mają liczbę Erdösa równą 1. Osoby, które współpracowały z osobami, które współpracowały z Erdösem, mają liczbę Erdösa równą 2. I tak dalej.

Marzeniem więc wielu młodych matematyków jest oczywiście mieć liczbę Erdösa jak najwyższą. Los jednak spłatał im figla – czy też raczej była to naturalna kolej rzeczy – Paul Erdös zmarł ponad 10 lat temu, w 1996 roku w wieku ponad 80 lat.

Liczby Erdösa były też czasem przypisywane osobom nie parającym się matematykom – najdziwniejszym przypadkiem będzie zapewne pewien arcyinteligentny koń, posiadający liczbę Erdösa 3 (to niższa niż moja!).

La Galerie Immanence (ici) exposait de la jeune photographie hongroise jusqu’au 28 mars (je suis un peu en retard). Cette galerie est située face au musée du Montparnasse (j’en ai parlé ici).

Ce n’est que maintenant (plus d’un mois après donc) que je découvre le titre “officiel” de l’exposition (Regardez-moi, i’m back) alors même que je n’avais pas remarqué sur les photos que tous les personnages étaient… de dos. Comme quoi. Ceci dit j’ai une excuse puisqu’il n’y avait pas que des portraits et des personnes de représentées. A posteriori, la récole est finalement mince : peu d’auteurs disposent d’un site web et il est difficile de cerner le travail de chacun.

D’ailleurs, je ne comprends pas qu’en 2009, de jeunes photographes qui prétendent à être vus au-delà des frontières nationales ne disposent pas d’un site web complet et à jour (je parle pas d’une page facebook ou d’un blog – bienvenus mais vraiment pas prioritaires) , au moins partiellement traduit en anglais et correctement référencé. Bref.

Gabor Erdos nous montre un intérieur banal et avec le nom qu’il porte (très répandu), inutile de chercher un site web pour en trouver plus.  Étant ignare en hongrois, j’ai aussi laissé tomber les signes diacritiques en espérant ne pas être lynché. Roland Biro (25 ans seulement) montrait, d’après mes notes, “un diasec avec une auréole”  comme on trouve sur son site (ici et notamment là) mais je ne retrouve pas exactement de quoi il s’agissait. En tout cas, son travail est assez surprenant : un peu d’icône, un peu de surréalisme, un peu de poésie et du doré partout.

Mate Moro montre des filles en reflets dans de petits formats. Son blog (ici) semble abandonné depuis novembre 2008. Istvan Pok montre des femmes en robe de dos grâce à des Polaroïds transférés sur papier aquarelle ce qui donne une tonalité particulière à son travail mais là encore, pas de site web.

Gabor Arion Kudasz nous montre  des “extérieurs nuit” avec un type et une valise et une de ses photos figure sur le site de la galerie (ci-dessous). Pour le reste, son site web est reporté par Avast comme hébergeant un cheval de troie (!) et son travail est vendu chez un marchand de poster (Coin Jaune). Pas de commentaire.

Tamas Dezso (qui a un site web ici mais pas avec la photo montrée) présentait un triptyque : une fenêtre de HLM avec un personnage devant. Gabor Kasza (son site ici hélas en flash) montre un genou (un coude) abimé qui se confond avec le fond (série ici). Balazs Simonyi présentait 4 vues d’un dos masculin. Son site (ici) ne marche pas. Peter Puklus (site ici)  montrait trois  scenes de chambres aux couleurs très 70′.

Istvan Krajnik (son site très bien fait, ici) montre une grosse dame nue de dos à table.

Le travail de Krisztina Erdei est le plus intéressant à mon goût (son travail est visible ici hélas en flash et là) avec 5 amusantes photos un peu à la Parr.

As you probably know, a prime number is a natural number like 2, 3, or 53, which is only evenly divisible by two numbes:  1 and itself.  The idea is that a prime number is a number which cannot be written as the product of two smaller numbers.  They act like the “atoms” of the natural numbers.  The Fundamental Theorem of Arithmetic says that every natural number can be written as a product of prime numbers in one and only one way.

It is easy to see that every number can be written as the product of primes by the following algorithm.  Take your number, if it’s prime you are done.  If not, then it can be written as the product of two smaller numbers.  Now repeat by applying the algorithm to the two smaller numbers.  You keep breaking down the factors into smaller and smaller pieces until you can’t anymore.  Once you can’t break any of the factors down further, then it means you have reached prime numbers.  The harder part of the theorem is to show that no matter how you do this, you end up with the same prime numbers.

Once you start thinking about prime numbers, you pretty quickly ask “How many are there?”  The short answer is “Infinitely many.”  The long answer is that there is a lot of different proofs for why there are infinitely many primes.  Why does anybody need more than one proof?  Because different proofs shed light on different aspects of the prime numbers and help us to better understand them.  Let’s look at a some of the different proofs of the fact that there is infinitely many primes below.

1. The oldest known proof is from Euclid’s Elements from around 300 BC.  The proof is by contradiction:  Say that there is only a finite number of primes.  Let’s write them in a list from smallest to biggest:

where N is the number of prime numbers.  Consider the number you get by multiplying them all together and adding 1:

If E is not divisible by any number other than itself and 1, then it is prime.  But E is bigger than any of the numbers in our list, so E would be a prime which is not in our list.  This contradicts our assumption that the list contains all primes.  So E is not prime.  This means we can write it as the product of two smaller numbers and, as we saw above, this means there is a prime number which divides E evenly.  Now the primes from our list do not divide E evenly since when you divide you get a remainder of 1.  So we have found a prime which is not on our list.  But this again contradicts our assumption that the list contained all primes.  Where did we go wrong?  By assuming that there was finitely many primes so that we could make the list in the first place!  Therefore there must be infinitely many primes.

This is a very nice and non-technical proof, and makes use of the fact that any number is either prime or is evenly divisible by a prime, which would suggest the Fundamental Theorem of Arithmetic to someone who didn’t already know about it.

Euclid in a painting by Raphael

2. Perhaps the newest proof uses the Green – Tao Theorem which we discussed here.  The Green – Tao Theorem was proven a few years ago and says that there are arbitrarily long arithmetic sequences in the primes.  The fact that there is infinitely many primes is a simple corollary of the theorem (we’ll let you think about why there has to be infinitely many primes if the Green – Tao Theorem is true).

It is cutting butter with a chainsaw to use the Green – Tao Theorem to prove that there is infinitely many primes, but this proof does bring a person’s attention to the arithmetic sequences in the primes.

Warning: We don’t know the details of the Green – Tao Theorem, and it may well use the fact that there is infinitely many primes in the proof itself.  Of course you are not allowed to use a fact to prove that same fact, so this proof might not be allowed!

3. The next proof uses the Mersenne numbers which we discussed here.  If there is finitely many prime numbers, then there is a largest prime number.  Let’s call it P.  Then consider the Mersenne number

Observe that M is bigger than P.  If M is prime, then we have contradicted the assumption that P is the largest prime.  If M is not prime, we know that there is a prime number which divides M evenly.  Let’s call that prime q.  Since q divides M evenly, that means that equals 1 in the group of nonzero elements of with multiplication as the binary operation.  This means 2 has order P in this group.  But this group has q – 1 elements.  But by Legrange’s Theorem from group theory,  the order of an element of a group has to evenly divide the size of the group.  That means that P divides q-1 evenly.  Which means  .  But that means P is smaller than q, contradicting our assumption that P was the biggest prime.  Where did we go wrong?  By assuming there was a biggest prime; that is, assuming that there was finitely many prime numbers.

The nice thing about this proof is that it points out the usefulness of the Mersenne Primes, using the size of natural numbers in proofs, and how you can use seemingly unrelated mathematics (like Legrange’s theorem) to prove the result you want.

Prime phone numbers in Toothpaste for Dinner

4. Another proof comes from a famous theorem called Bertrand’s Postulate. Bertrand’s postulate says that for any natural number n, there is a prime number between n and 2n.  It was conjectured by Bertrand after he checked all the numbers up to 3 million!  It was first proven by Chebyshev, then new proofs were given by Ramanujan and then by Erdos when he was only 19 years old!

This leads us to a famous quote by Nathan Fine pointed out to us by Dr. Forester:

Chebyshev said it, but I’ll say it again; There’s always a prime between n and 2n.

It’s easy to see that Bertrand’s Postulate implies there is infinitely many prime numbers.  But it also gives you some feeling for how spaced out the primes are.  It also suggests that the following result might be true:  for any positive integer k, there is a natural number N such that for all n > N, there are at least k primes between n and 2n.  This is true and was proven both by Ramanujan and Erdos.

A variation suggested by Bertrand’s Postulate is the still open Legendre’s Conjecture: Is there always a prime number between and ?  Nobody knows!

5. One of our favorite proofs that there is infinitely many primes is a consequence of the following Calc III type theorem.  It was first proven by Euler and then a different proof was given Erdos.  Let

be the list of prime numbers.  Consider the series

Euler and Erdos proved that this series diverges!

Of course this implies there is infinitely many primes since if there was only finitely many, then the above series would be a finite sum and so would converge.  But more than that, we know that the series

converges, so roughly speaking we know that there is more primes then there is perfect squares!  So this theorem tells you something about how “many” primes there are.  That is, there is infinitely many primes and infinitely many perfect squares, but there is “more” primes.   Similar comparisions can be made with your other favorite converging or diverging series.

Euler on a German Stamp. Extra credit if you recognise the formula (Hint: Play-doh!).

6. There are many, many other proofs out there.  For example, the fact that there are infinitely many primes is an easy consequence of one of the most important open questions in mathematics, the Riemann hypothesis (By the way, in case you need to make some spare change, proving/disproving the Riemann hypothesis is one of the million dollar Millenium Prize Problems of the Clay Institute).

Amazingly, there is a topology proof that there is infinitely many primes!  It was given by Hillel Furstenberg and published in the Math Monthly while he was an undergraduate!  See here for the proof.

N is a Number: A Portrait of Paul Erdös

I watched a film the other day I hadn’t seen in a while – N is a Number: A Portrait of Paul Erdös. It’s great! Ok, so it’s more of a documentary than a movie, but it’s very interesting. Paul Erdös was a famous mathematician and already down in the books before passing away (just 3 years after the documentary was released). He has published more research papers than any other mathematician in history with close to 1500 publications. He worked on problems in combinatorics, graph theory, and many other fields, especially number theory. An idiosyncratic character, Erdös often worked simultaneously with dozens of collaborators across the globe; all this while being homeless and unemployed. He didn’t have a fixed address and all the money he made came from lectures and talks at the universities he visited.
I think what’s most fascinating about this film is that you get the chance to see Erdös on tape. A documentary about a famous figure featuring that person is not as rare in music, literature, film, or painting as it is in mathematics. The only other film I am aware meets this criteria is Clouds are Not Spheres about the life of Benoît Mandelbrot, commonly known as the founder of fractal geometry. Guess what movie to watch is next on my list =)

BARCELONA – MADRID

Sábado 13/12/08 (22:00)

Para añadir más pique si cabe al partido. En principio se habló de poner 3 € pero seguramente será mas que nada honorífica debido a la recaudación del dinero.

Se irá añadiendo los resultados de la gente (hay que decir resultado y goles) que vayais poniendo en los comentarios.

—————————————————————————————-

• Gelo: (1-3) (Valdés, 2 Higuaín, Ramos)
• Bejisa: (2-0) (2 Messi)
• Natxo (telemática): (2-2) (Xavi, Messi, Higuaín, VDV)
• Jaime: (3-1) (Messi, Henry, Bojan, Raul)
• Luci: (4-2) (Etoo, 2 Messi, Xavi, Raul, Higuaín)
• Charlie: (3-0) (Messi, Xavi y el “hermano”)
• Pablo: (4-1) (2 Etoo, Xavi, Titi, Pipita)
• Wiky: (1-2) (Messi, Raul, Pipita)
• Maria: (1-2) (Messi, VDV, Higuain)
• Lolo: (3-1) (2 Etoo, Messi, Robben) Muy bien Lolo, Robben sancionado
• Luis: (¿Quien juega?… Pos Cazorla)
• Guti: (5-1) (2 Messi, 2 Etoo, Xavi, Raul)
• Petxo: (2-o) (Etoo, Puyol)
• Arce(¿?): (7-1) (2Messi, Etoo,Titi, Xavi, Alves, Gud, Raul)
• Kike: (0-1) (Pipita)
• Ivo: (4-0) (2 Messi, Etoo, Gudjhonsen)

—————————————————————————————-

]]> http://quomodocumque.wordpress.com/2008/12/08/erdos-20/ Mon, 08 Dec 2008 14:45:35 +0000 JSE http://quomodocumque.wordpress.com/2008/12/08/erdos-20/

Will the next Erdös be someone who hangs around at home, reads a lot of math blogs, and posts solutions to open problems in the comments?

Paul Erdos

This is short notice, but there will be a talk in the OU Graduate Students’ Pizza Seminar this Monday, October 24th in PHSC 1105 at 5pm. The talk is about the mathematics of Paul Erdos.  The material will be accessible to everyone (not even calculus is used!).

The details are below.

If nothing else, check out the above link to Wikipedia’s article on Erdos.  He’s one of the most interesting people you can imagine!

Speaker: Jon Kujawa, University of Oklahoma

Title: Solved and Unsolved Problems of Paul Erdos

Abstract: Paul Erdos was one of the most prolific mathematicians of all time (1587 papers with at least 511 collaborators!).  He worked in (and invented) a large number of areas of mathematics; including graph theory, probabilistic combinatorics, number theory, extremal combinatorics, and geometry.  Famous for his ability to solve problems, he was even more known for his ability to pose problems which cut right to the heart of a mathematical question.  We will discuss some of these problems, both solved and unsolved.  Cash prizes will be offered!

As always, free pizza will be provided and all are welcome!

[From Erdős to Kiev] Ada sebuah papan kotak-kotak berukuran 50×50 dan sebuah dadu yang sisinya sama besar dengan ukuran satu kotak. Mula-mula dadu itu ditaruh di sudut kiri bawah. Kemudian dadu digelindingkan sampai ke sudut kanan atas. Dalam setiap langkah, dadu hanya boleh bergerak ke kanan atau ke atas. Seperti biasa, dadu diberi nomor 1 sampai 6, dengan syarat dua sisi berlawanan berjumlah 7. Setiap kali dadu menempati satu kotak, nomor yang terlihat di sisi atas dadu itu akan tercetak di kotak itu. Berapa kemungkinan terbesar dan terkecil dari jumlah semua nomor pada kotak-kotak itu?

Solusi
Anggaplah pertama-tama sisi dadu yang muncul bernomor x. Agar muncul x lagi, pasti muncul 7-x dulu sebelumnya. Jadi di antara terjadinya x dua kali, pasti ada 7-x. Begitu juga sebaliknya, di antara dua kali kemunculan 7-x, pasti ada x. Jadi pasangan (x, 7-x) terus menerus muncul bergantian.

Anggaplah x muncul lebih dahulu sebelum 7-x. Pada akhirnya, ada kemungkinan bahwa 7-x tidak muncul lagi, dan x yang muncul terakhir. Ini kita sebut pasangan tidak lengkap. Karena ada tiga pasangan (x, 7-x), yaitu (1,6), (2,5), (3,4), maka banyaknya pasangan tidak lengkap maksimum adalah 3.

Perhatikan bahwa kita perlu 99 langkah. Setiap pasangan lengkap memakai dua langkah. Jadi banyaknya pasangan tidak lengkap tidak mungkin ada dua, karena menyebabkan banyaknya langkah bilangan genap, bukan 99. Maka banyaknya pasangan tidak lengkap adalah 1 atau 3, sehingga pasangan lengkap ada 49 atau 48.

Anggaplah ada 1 pasangan tidak lengkap dan 49 pasangan lengkap. Setiap pasangan lengkap memiliki jumlah 7, sehingga totalnya 343. Pasangan tidak lengkap terakhir bisa memiliki nilai 1, 2, 3, 4, 5, atau 6. Jadi total jumlah semua nomornya adalah 344, 345, 346, 347, 348, atau 349.

Sekarang anggaplah ada 3 pasangan tidak lengkap dan 48 pasangan lengkap. Pasangan lengkap memiliki jumlah 336. Pasangan tidak lengkapnya adalah di antara (1,6), (2,5), (3,4). Maka jumlah maksimumnya adalah 336+6+5+4=351, dan jumlah minimumnya 336+1+2+3=342.

Mudah dilihat bahwa nilai minimum dan maksimum ini dapat dicapai. Jadi nilai maksimum adalah 351 dan nilai minimum adalah 342.

Paul Erdös a établi un lien indiscutable entre le café et la Science : Un mathématicien est une machine à transformer le café en théorèmes. . D’autres auteurs ont reconnu les vertus du précieux nectar dans des domaines intellectuels variés, notamment : “Le café, c’est amer et ça donne envie de faire pipi, mais ça empêche de dormir alors on peut programmer plus longtemps” (moi).

Je suis donc très honoré que ce blog ait été accepté comme membre du C@fé des Sciences, la communauté des blogs scientifiques en français. Le C@fé est accessible depuis Dr. Goulu grâce à une icône sous “Sites liés” dans la colonne de gauche, et les titres des derniers articles des membres de la communauté sont repris dans un flux RSS juste en dessous.

Revenons à Paul Erdös pour relever le niveau de ce billet. Ce mathématicien hongrois s’intéressait notamment aux graphes et a publié plus de 1500 articles en près de 60 ans de carrière. Il a très donc logiquement proposé de représenter la collaboration entre auteurs scientifiques (en mathématique essentiellement) sous forme d’un graphe qui permet de définir le “nombre d’Erdös” de chaque chercheur :

• Erdös lui même a un nombre d’Erdös = 0
• les plus de 504 qui ont co-publié au moins 1 article avec Erdös ont un nombre = 1
• les 6593 cosignataires d’articles écrits avec ces 504 auteurs reçoivent un nombre d’Erdös = 2
• et ainsi de suite. Le “Erdös Number Project” possède actuellement un graphe de 1.9 millions d’articles publiés par 420′000 auteurs, ce qui permet d’en tirer toutes sortes de statistiques intéressantes sur la publication des mathématiciens.

Des études de ce type permettent de synthétiser et visualiser la collaboration comme dans la magnifique Carte des Sciences, mais aussi pour étudier les réseaux sociaux sur le Web 2.0, ou le mécanisme de publication sur les blogs excellement étudié dans “Généalogie (in)signifiante de mèmes” par TomRoud.com.

C’est un membre du C@fé, la boucle est bouclée.