Project B08 - The Allure of the Primes

Euclid may have been one of the first to be interested in the prime numbers, but over the last two millennia, many people have joined his ranks. Since every whole number greater than one can be decomposed into prime factors, the prime numbers are, in a certain sense, the building blocks of our numeration system. People wishing to learn more about our numeration system must learn more about the prime numbers.

Who is Euclid? Write a paragraph about him and some of his accomplishments.
Other than the reason suggested above, why do people study prime numbers and search for large prime numbers?
One way to find all the primes less than a given number is to use the Sieve of Eratosthenes, named after the Greek mathematician Eratosthenes. Use this method to find all the primes less than 200. (See the attached sheet.)
My favorite prime number in your list is 29, because that's how old I was before I turned 30. What is your favorite prime less than 200 and why?
A number whose digits are ones (and only ones) is called a repunit. How many prime repunits are less than 200? Find an infinite sequence of composite repunits.
What is a Mersenne number? What is the smallest composite Mersenne number? What is the greatest prime Mersenne number? (There are two different accepted definitions of Mersenne numbers. Try to find both definitions and work this problem with each.)
What is the Twin Prime Conjecture?

Harvey Dubner and collaborators have recently found the longest known arithmetic progression of palindromic primes:

742950290870000078092059247
742950290871010178092059247
742950290872020278092059247
742950290873030378092059247
742950290874040478092059247
742950290875050578092059247
742950290876060678092059247
742950290877070778092059247
742950290878080878092059247
742950290879090978092059247

Explain in layman's terms what these researchers have found.

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On 13 Nov 1999, 17:12.