Noam David Elkies is an American mathematician and professor of mathematics at Harvard University. At the age of 26, he became the youngest professor to receive tenure at Harvard, he is a chess master and an accomplished chess composer. Elkies attended Stuyvesant High School in New York City for three years before graduating in 1982. In 1981, at age 14, he was awarded a gold medal at the 22nd International Mathematical Olympiad, receiving a perfect score of 42, he went on to Columbia University, where he won the Putnam competition at the age of sixteen years and four months, making him one of the youngest Putnam Fellows in history. He was a Putnam Fellow two more times during his undergraduate years, he earned his Ph. D. in 1987 under the supervision of Benedict Gross and Barry Mazur at Harvard University. From 1987 to 1990 he was a junior fellow of the Harvard Society of Fellows. In 1987, he proved that an elliptic curve over the rational numbers is supersingular at infinitely many primes. In 1988, he found a counterexample to Euler's sum of powers conjecture for fourth powers.
His work on these and other problems won him recognition and a position as an associate professor at Harvard in 1990. In 1993, he was made a full, tenured professor at the age of 26; this made him the youngest full professor in the history of Harvard. Along with A. O. L. Atkin he extended Schoof's algorithm to create the Schoof–Elkies–Atkin algorithm. Elkies studies the connections between music and mathematics, he has discovered many new patterns in Conway's Game of Life and has studied the mathematics of still life patterns in that cellular automaton rule. Elkies is an associate of Harvard's Lowell House, he plays the piano for Harvard Glee Club. In an article Jameson N. Marvin the director of the Glee Club compares him to Bach or a Mozart citing "his gifted musicality, superior musicianship and sight-reading ability". Elkies is a solver of chess problems, he holds the title of National Master from the United States Chess Federation, but he no longer plays competitively. In 1994 he was an invited speaker at the International Congress of Mathematicians in Zurich.
In 2004 he received the Levi L. Conant Prize. In 2017 he was elected to the National Academy of Sciences. Personal site of Noam Elkies at Harvard University Endgame Explorations – an 11-part series of articles by Noam Elkies in Chess Horizons Noam Elkies on LifeWiki
Fermat's Last Theorem
In number theory Fermat's Last Theorem states that no three positive integers a, b, c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have an infinite number of solutions; the proposition was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica. However, there were first doubts about it since the publication was done by his son without his consent, after Fermat's death. After 358 years of effort by mathematicians, the first successful proof was released in 1994 by Andrew Wiles, formally published in 1995, it proved much of the modularity theorem and opened up entire new approaches to numerous other problems and mathematically powerful modularity lifting techniques. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century, it is among the most notable theorems in the history of mathematics and prior to its proof was in the Guinness Book of World Records as the "most difficult mathematical problem" in part because the theorem has the largest number of unsuccessful proofs.
The Pythagorean equation, x2 + y2 = z2, has an infinite number of positive integer solutions for x, y, z. Around 1637, Fermat wrote in the margin of a book that the more general equation an + bn = cn had no solutions in positive integers if n is an integer greater than 2. Although he claimed to have a general proof of his conjecture, Fermat left no details of his proof, no proof by him has been found, his claim was discovered some 30 years after his death. This claim, which came to be known as Fermat's Last Theorem, stood unsolved for the next three and a half centuries; the claim became one of the most notable unsolved problems of mathematics. Attempts to prove it prompted substantial development in number theory, over time Fermat's Last Theorem gained prominence as an unsolved problem in mathematics; the special case n = 4 - proved by Fermat himself - is sufficient to establish that if the theorem is false for some exponent n, not a prime number, it must be false for some smaller n, so only prime values of n need further investigation.
Over the next two centuries, the conjecture was proved for only the primes 3, 5, 7, although Sophie Germain innovated and proved an approach, relevant to an entire class of primes. In the mid-19th century, Ernst Kummer extended this and proved the theorem for all regular primes, leaving irregular primes to be analyzed individually. Building on Kummer's work and using sophisticated computer studies, other mathematicians were able to extend the proof to cover all prime exponents up to four million, but a proof for all exponents was inaccessible. Separately, around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama suspected a link might exist between elliptic curves and modular forms, two different areas of mathematics. Known at the time as the Taniyama–Shimura–Weil conjecture, as the modularity theorem, it stood on its own, with no apparent connection to Fermat's Last Theorem, it was seen as significant and important in its own right, but was considered inaccessible to proof. In 1984, Gerhard Frey noticed an apparent link between these two unrelated and unsolved problems.
An outline suggesting this could be proved was given by Frey. The full proof that the two problems were linked was accomplished in 1986 by Ken Ribet, building on a partial proof by Jean-Pierre Serre, who proved all but one part known as the "epsilon conjecture"; these papers by Frey and Ribet showed that if the Modularity Theorem could be proven for at least the semi-stable class of elliptic curves, a proof of Fermat's Last Theorem would follow automatically. The connection is described below: any solution that could contradict Fermat's Last Theorem could be used to contradict the Modularity Theorem. So if the modularity theorem were found to be true by definition no solution contradicting Fermat's Last Theorem could exist, which would therefore have to be true as well. Although both problems were daunting and considered to be "completely inaccessible" to proof at the time, this was the first suggestion of a route by which Fermat's Last Theorem could be extended and proved for all numbers, not just some numbers.
Important for researchers choosing a research topic was the fact that unlike Fermat's Last Theorem the Modularity Theorem was a major active research area for which a proof was desired and not just a historical oddity, so time spent working on it could be justified professionally. However, general opinion was that this showed the impracticality of proving the Taniyama–Shimura conjecture. Mathematician John Coates' quoted reaction was a common one: "I myself was sceptical that the beautiful link between Fermat’s Last Theorem and the Taniyama–Shimura conjecture would lead to anything, because I must confess I did not think that the Taniyama–Shimura conjecture was accessible to proof. Beautiful though this problem was, it seemed impossible to prove. I must confess I thought I wouldn’t see it proved in my lifetime." On hearing that Ribet had proven Frey's li
Leonhard Euler was a Swiss mathematician, astronomer and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while making pioneering contributions to several branches such as topology and analytic number theory. He introduced much of the modern mathematical terminology and notation for mathematical analysis, such as the notion of a mathematical function, he is known for his work in mechanics, fluid dynamics, optics and music theory. Euler was one of the most eminent mathematicians of the 18th century and is held to be one of the greatest in history, he is widely considered to be the most prolific mathematician of all time. His collected works fill more than anybody in the field, he spent most of his adult life in Saint Petersburg, in Berlin the capital of Prussia. A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all." Leonhard Euler was born on 15 April 1707, in Basel, Switzerland to Paul III Euler, a pastor of the Reformed Church, Marguerite née Brucker, a pastor's daughter.
He had two younger sisters: Anna Maria and Maria Magdalena, a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, where Euler spent most of his childhood. Paul Euler was a friend of the Bernoulli family. Euler's formal education started in Basel. In 1720, aged thirteen, he enrolled at the University of Basel, in 1723, he received a Master of Philosophy with a dissertation that compared the philosophies of Descartes and Newton. During that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who discovered his new pupil's incredible talent for mathematics. At that time Euler's main studies included theology and Hebrew at his father's urging in order to become a pastor, but Bernoulli convinced his father that Leonhard was destined to become a great mathematician. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono. At that time, he was unsuccessfully attempting to obtain a position at the University of Basel.
In 1727, he first entered the Paris Academy Prize Problem competition. Pierre Bouguer, who became known as "the father of naval architecture", won and Euler took second place. Euler won this annual prize twelve times. Around this time Johann Bernoulli's two sons and Nicolaus, were working at the Imperial Russian Academy of Sciences in Saint Petersburg. On 31 July 1726, Nicolaus died of appendicitis after spending less than a year in Russia, when Daniel assumed his brother's position in the mathematics/physics division, he recommended that the post in physiology that he had vacated be filled by his friend Euler. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to Saint Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel. Euler arrived in Saint Petersburg on 17 May 1727, he was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he worked in close collaboration.
Euler settled into life in Saint Petersburg. He took on an additional job as a medic in the Russian Navy; the Academy at Saint Petersburg, established by Peter the Great, was intended to improve education in Russia and to close the scientific gap with Western Europe. As a result, it was made attractive to foreign scholars like Euler; the academy possessed ample financial resources and a comprehensive library drawn from the private libraries of Peter himself and of the nobility. Few students were enrolled in the academy in order to lessen the faculty's teaching burden, the academy emphasized research and offered to its faculty both the time and the freedom to pursue scientific questions; the Academy's benefactress, Catherine I, who had continued the progressive policies of her late husband, died on the day of Euler's arrival. The Russian nobility gained power upon the ascension of the twelve-year-old Peter II; the nobility was suspicious of the academy's foreign scientists, thus cut funding and caused other difficulties for Euler and his colleagues.
Conditions improved after the death of Peter II, Euler swiftly rose through the ranks in the academy and was made a professor of physics in 1731. Two years Daniel Bernoulli, fed up with the censorship and hostility he faced at Saint Petersburg, left for Basel. Euler succeeded him as the head of the mathematics department. On 7 January 1734, he married Katharina Gsell, a daughter of Georg Gsell, a painter from the Academy Gymnasium; the young couple bought a house by the Neva River. Of their thirteen children, only five survived childhood. Concerned about the continuing turmoil in Russia, Euler left St. Petersburg on 19 June 1741 to take up a post at the Berlin Academy, which he had been offered by Frederick the Great of Prussia, he lived for 25 years in Berlin. In Berlin, he published the two works for which he would become most renowned: the Introductio in analysin infinitorum, a text on functions published in 1748, the Institutiones calculi differentialis, published in 1755 on differential calculus.
Srinivasa Ramanujan FRS was an Indian mathematician who lived during the British Rule in India. Though he had no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, continued fractions, including solutions to mathematical problems considered to be unsolvable. Ramanujan developed his own mathematical research in isolation: "He tried to interest the leading professional mathematicians in his work, but failed for the most part. What he had to show them was too novel, too unfamiliar, additionally presented in unusual ways. Seeking mathematicians who could better understand his work, in 1913 he began a postal partnership with the English mathematician G. H. Hardy at the University of Cambridge, England. Recognizing the extraordinary work sent to him as samples, Hardy arranged travel for Ramanujan to Cambridge. In his notes, Ramanujan had produced groundbreaking new theorems, including some that Hardy stated had "defeated completely", in addition to rediscovering proven but advanced results.
During his short life, Ramanujan independently compiled nearly 3,900 results. Many were novel. Nearly all his claims have now been proven correct; the Ramanujan Journal, a peer-reviewed scientific journal, was established to publish work in all areas of mathematics influenced by Ramanujan, his notebooks—containing summaries of his published and unpublished results—have been analyzed and studied for decades since his death as a source of new mathematical ideas. As late as 2011 and again in 2012, researchers continued to discover that mere comments in his writings about "simple properties" and "similar outputs" for certain findings were themselves profound and subtle number theory results that remained unsuspected until nearly a century after his death, he became one of the youngest Fellows of the Royal Society and only the second Indian member, the first Indian to be elected a Fellow of Trinity College, Cambridge. Of his original letters, Hardy stated that a single look was enough to show they could only have been written by a mathematician of the highest calibre, comparing Ramanujan to other mathematical geniuses such as Euler and Jacobi.
In 1919, ill health—now believed to have been hepatic amoebiasis —compelled Ramanujan's return to India, where he died in 1920 at the age of 32. His last letters to Hardy, written January 1920, show that he was still continuing to produce new mathematical ideas and theorems, his "lost notebook", containing discoveries from the last year of his life, caused great excitement among mathematicians when it was rediscovered in 1976. A religious Hindu, Ramanujan credited his substantial mathematical capacities to divinity, stated that the mathematical knowledge he displayed was revealed to him by his family goddess. "An equation for me has no meaning," he once said, "unless it expresses a thought of God." Ramanujan was born on 22 December 1887 into a Tamil Brahmin Iyengar family in Erode, Madras Presidency, at the residence of his maternal grandparents. His father, Kuppuswamy Srinivasa Iyengar from Thanjavur district, worked as a clerk in a sari shop, his mother, was a housewife and sang at a local temple.
They lived in a small traditional home on Sarangapani Sannidhi Street in the town of Kumbakonam. The family home is now a museum; when Ramanujan was a year and a half old, his mother gave birth to a son, who died less than three months later. In December 1889, Ramanujan contracted smallpox, though he recovered, unlike 4,000 others who would die in a bad year in the Thanjavur district around this time, he moved with his mother near Madras. His mother gave birth to two more children, in 1891 and 1894, both failing to reach their first birthdays. On 1 October 1892, Ramanujan was enrolled at the local school. After his maternal grandfather lost his job as a court official in Kanchipuram and his mother moved back to Kumbakonam and he was enrolled in the Kangayan Primary School; when his paternal grandfather died, he was sent back to his maternal grandparents living in Madras. He did not like school in Madras, tried to avoid attending, his family enlisted a local constable to make sure. Within six months, Ramanujan was back in Kumbakonam.
Since Ramanujan's father was at work most of the day, his mother took care of the boy as a child. He had a close relationship with her. From her, he learned about tradition and puranas, he learned to sing religious songs, to attend pujas at the temple, to maintain particular eating habits—all of which are part of Brahmin culture. At the Kangayan Primary School, Ramanujan performed well. Just before turning 10, in November 1897, he passed his primary examinations in English, Tamil and arithmetic with the best scores in the district; that year, Ramanujan entered Town Higher Secondary School, where he encountered formal mathematics for the first time. By age 11, he had exhausted the mathematical knowledge of two college students who were lodgers at his home, he was lent a book by S. L. Loney on advanced trigonometry, he mastered this by the age of 13. By 14, he was receiving merit certificates and acade
In mathematical programming and polyhedral combinatorics, the Hirsch conjecture is the statement that the edge-vertex graph of an n-facet polytope in d-dimensional Euclidean space has diameter no more than n − d. That is, any two vertices of the polytope must be connected to each other by a path of length at most n − d; the conjecture was first put forth in a letter by Warren M. Hirsch to George B. Dantzig in 1957 and was motivated by the analysis of the simplex method in linear programming, as the diameter of a polytope provides a lower bound on the number of steps needed by the simplex method; the conjecture is now known to be false in general. The Hirsch conjecture was proven for d < 4 and for various special cases, while the best known upper bounds on the diameter are only sub-exponential in n and d. After more than fifty years, a counter-example was announced in May 2010 by Francisco Santos Leal, from the University of Cantabria; the result was presented at the conference 100 Years in Seattle: the mathematics of Klee and Grünbaum and appeared in Annals of Mathematics.
The paper presented a 43-dimensional polytope of 86 facets with a diameter of more than 43. The counterexample has no direct consequences for the analysis of the simplex method, as it does not rule out the possibility of a larger but still linear or polynomial number of steps. Various equivalent formulations of the problem had been given, such as the d-step conjecture, which states that the diameter of any 2d-facet polytope in d-dimensional Euclidean space is no more than d; the graph of a convex polytope P is any graph whose vertices are in bijection with the vertices of P in such a way that any two vertices of the graph are joined by an edge if and only if the two corresponding vertices of P are joined by an edge of the polytope. The diameter of P, denoted δ, is the diameter of any one of its graphs; these definitions are well-defined since any two graphs of the same polytope must be isomorphic as graphs. We may state the Hirsch conjecture as follows: Conjecture Let P be a d-dimensional convex polytope with n facets.
Δ ≤ n − d. For example, a cube in three dimensions has six facets; the Hirsch conjecture indicates that the diameter of this cube cannot be greater than three. Accepting the conjecture would imply that any two vertices of the cube may be connected by a path from vertex to vertex using, at most, three steps. For all polytopes of dimension at least 8, this bound is optimal. In other words, for nearly all cases, the conjecture provides the minimum number of steps needed to join any two vertices of a polytope by a path along its edges. Since the simplex method operates by constructing a path from some vertex of the feasible region to an optimal point, the Hirsch conjecture would provide a lower bound needed for the simplex method to terminate in the worst case scenario; the Hirsch conjecture is a special case of the polynomial Hirsch conjecture, which claims that there exists some positive integer k such that, for all polytopes P, δ = O, where n is the number of facets of P. The Hirsch conjecture has been proven true for a number of cases.
For example, any polytope with dimension 3 or lower satisfies the conjecture. Any d-dimensional polytope with n facets. Other attempts to solve the conjecture manifested out of a desire to formulate a different problem whose solution would imply the Hirsch conjecture. One example of particular importance is the d-step conjecture, a relaxation of the Hirsch conjecture, shown to be equivalent to it. Theorem The following statements are equivalent: δ ≤ n − d for all d-dimensional polytopes P with n facets. Δ ≤ d for all d-dimensional polytopes P with 2d facets. In other words, in order to prove or disprove the Hirsch conjecture, one only needs to consider polytopes with twice as many facets as its dimension. Another significant relaxation is that the Hirsch conjecture holds for all polytopes if and only if it holds for all simple polytopes; the Hirsch conjecture is not true in all cases, as shown by Francisco Santos in 2011. Santos' explicit construction of a counterexample comes both from the fact that the conjecture may be relaxed to only consider simple polytopes, from equivalence between the Hirsch and d-step conjectures.
In particular, Santos produces his counterexample by examining a particular class of polytopes called spindles. Definition A d-spindle is a d-dimensional polytope P for which there exist a pair of distinct vertices such that every facet of P contains one of these two vertices; the length of the shortest path between these two vertices is c
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
The CDC 6600 was the flagship of the 6000 series of mainframe computer systems manufactured by Control Data Corporation. Considered to be the first successful supercomputer, it outperformed the industry's prior recordholder, the IBM 7030 Stretch, by a factor of three. With performance of up to three megaFLOPS, the CDC 6600 was the world's fastest computer from 1964 to 1969, when it relinquished that status to its successor, the CDC 7600; the first CDC 6600's were delivered in 1965 to Los Alamos. They became a must-have system in scientific and mathematical computing circles, with systems being delivered to Courant Institute of Mathematical Sciences, CERN, the Lawrence Radiation Laboratory, many others. 50 were delivered in total. A CDC 6600 is on display at the Computer History Museum in California; the only running CDC 6000 series machine has been restored by Living Computers: Museum + Labs. CDC's first products were based on the machines designed at ERA, which Seymour Cray had been asked to update after moving to CDC.
After an experimental machine known as the Little Character, in 1960 they delivered the CDC 1604, one of the first commercial transistor-based computers, one of the fastest machines on the market. Management was delighted, made plans for a new series of machines that were more tailored to business use. Cray was not interested in such a project, set himself the goal of producing a new machine that would be 50 times faster than the 1604; when asked to complete a detailed report on plans at one and five years into the future, he wrote back that his five-year goal was "to produce the largest computer in the world", "largest" at that time being synonymous with "fastest", that his one-year plan was "to be one-fifth of the way". Taking his core team to new offices nearby the original CDC headquarters, they started to experiment with higher quality versions of the "cheap" transistors Cray had used in the 1604. After much experimentation, they found that there was no way the germanium-based transistors could be run much faster than those used in the 1604.
The "business machine" that management had wanted, now forming as the CDC 3000 series, pushed them about as far as they could go. Cray decided the solution was to work with the then-new silicon-based transistors from Fairchild Semiconductor, which were just coming onto the market and offered improved switching performance. During this period, CDC grew from a startup to a large company and Cray became frustrated with what he saw as ridiculous management requirements. Things became more tense in 1962 when the new CDC 3600 started to near production quality, appeared to be what management wanted, when they wanted it. Cray told CDC's CEO, William Norris that something had to change, or he would leave the company. Norris felt he was too important to lose, gave Cray the green light to set up a new laboratory wherever he wanted. After a short search, Cray decided to return to his home town of Chippewa Falls, where he purchased a block of land and started up a new laboratory. Although this process introduced a lengthy delay in the design of his new machine, once in the new laboratory, without management interference, things started to progress quickly.
By this time, the new transistors were becoming quite reliable, modules built with them tended to work properly on the first try. The 6600 began to take form, with Cray working alongside Jim Thornton, system architect and "hidden genius" of the 6600. More than 100 CDC 6600s were sold over the machine's lifetime. Many of these went to various nuclear weapon-related laboratories, quite a few found their way into university computing laboratories. Cray turned his attention to its replacement, this time setting a goal of ten times the performance of the 6600, delivered as the CDC 7600; the CDC Cyber 70 and 170 computers were similar to the CDC 6600 in overall design and were nearly backwards compatible. The 6600 was three times faster than the IBM 7030 Stretch. Then-CEO Thomas Watson Jr. wrote a memo to his employees: "Last week, Control Data... announced the 6600 system. I understand that in the laboratory developing the system there are only 34 people including the janitor. Of these, 14 are engineers and 4 are programmers...
Contrasting this modest effort with our vast development activities, I fail to understand why we have lost our industry leadership position by letting someone else offer the world's most powerful computer." Cray's reply was sardonic: "It seems like Mr. Watson has answered his own question." Typical machines of the era used a single CPU to drive the entire system. A typical program would first load data into memory, process it, write it back out; this required the CPUs to be complex in order to handle the complete set of instructions they would be called on to perform. A complex CPU implied a large CPU, introducing signalling delays while information flowed between the individual modules making it up; these delays set a maximum upper limit on performance, the machine could only operate at a cycle speed that allowed the signals time to arrive at the next module. Cray took another approach. At the time, CPUs ran slower than the main memory to which they were attached. For instance, a processor might take 15 cycles to multiply two numbers, while each memory access took only one or two.
This meant. It was this idle time; the CDC 6600 used a simplified core processor that wa