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“Fibonacci Expansions and ‘F-adic’ integers,” The Fibonacci Quarterly, v. 34, 1996

“On the Sums of Digits of Fibonacci Numbers,” The Fibonacci Quarterly, v. 35, 1996..

A Modification of Shanks’ Baby-Step Giant-Step Algorithm,” Mathematics of Computation, v. 69, 2000

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History of Europe from 500 to 1900

Middle Ages
The Middle Ages is the period spanning from the fall of the Roman Empire around 500 AD to the beginning of the Italian Renaissance around 1450. It is often called the Dark Ages because much ancient knowledge was lost at the beginning of the Middle Ages, due to the invasions of warring factions from the north such as the Vandals, Visigoths, Goths, and later the Vikings. The Catholic church was the dominant power holding Europe together during most of this time. The height of the Middle Ages was during the 11th through 13th centuries, during which time a series of holy wars known as the Crusades were waged against the Muslims in the Middle East. During this period, much ancient knowledge was resurrected in Europe, including Aristotilian natural philosophy. Great cathedrals and universities were built throughout Europe during this period. Medieval Europe experienced a decline during the 14th and 15th centuries, however. The church began the corrupt practice of selling usuries and indulgences. The most devastating event in Europe was the Black Death (1347-1351), a pandemic which killed roughly one-third of the population. This period was also marred by the Hundred Years War between England and France from 1337 to 1451.

The Renaissance
The Renaissance was a period of revival in Europe which began in Italy around 1450. Renaissance means rebirth, as indicated by the rebirth of ancient Greek and Roman learning, resulting in a new humanist movement, which triggered advances in art and science. The Renaissance was characterized by great artists, including Leonardo Da Vinci and Michelangelo, as well as great inventions, such as the printing press and the pendulum clock. The Renaissance spread to Northern Europe around 1600 and lasted until around 1700.

The Protestant Reformation
This movement was begun by Martin Luther, who on October 31, 1517 posted his 95 Theses on the door of the Castle Church in Wittenburg, Germany, criticizing the sale of indulgences by the Catholic Church. Various divisions of Protestantism spread very quickly through Europe over the following years. Protestantism was often motivated by politics. For instance, King Henry VIII, who started out as a Catholic, ended up adopting Protestantism in 1534 so he could divorce his first wife, Catherine of Aragon.

The Age of Discovery
This age, from about 1420 to about 1620, was marked by great sea voyages in which new lands were discovered and new trade routes were established. The Portuguese carried out the first voyages of this era when Henry the Navigator discovered the Madeira Islands in 1419 and the Azores in 1427. In 1492, Christopher Columbus discovered America in an attempt to establish a trade route to India by sailing westward from Spain. Instead he reached the Caribbean Islands, which became known as the West Indies, and the native Americans as Indians. In 1519, Ferdinand Magellan led the first voyage around the world, which took three years and in which only one of four ships was able to complete the voyage.

The Scientific Revolution
The 16th and 17th centuries saw much scientific progress, beginning with Nicolaus Copernicus, who in 1543 suggested that the sun, not the earth, was the center of the universe. This was later verified by Galileo Galileii in 1609, following his invention of the telescope. Other great astronomers of this period included Tycho Brahe, Johannes Kepler, and Sir Isaac Newton. Scientific advances were also made in physics and anatomy and the scientific method of observation and experimentation was invented by Sir Francis Bacon.

The Age of Enlightenment
Many of the new scientific ideas led to the Age of Enlightenment, also known as the Age of Reason, in the 18th century. During this time, logic and reason were beginning to be applied to aesthetics, ethics, and government. The ideas of this period eventually led to the American and French Revolutions. Some enlightened thinkers included John Locke, Voltaire, and Jean-Jacques Rousseau.

The 19th Century
19th century Europe saw the Industrial Revolution and the rise of nationalism. Most European nations became constitutional monarchies during this period. The British Empire fluorished in the 19th century due to their victory over France in the Napoleonic Wars and the Industrial Revolution. The Congress of Vienna of 1815 led to a balance of power of the empires. Nationalistic tensions grew toward the end of the 19th century, however, eventually leading to World War I.