Pierre de Fermat (1601 - 1665). In the margin of his copy of Diophantus' Arithmetica, Fermat wrote “To divide a cube into two other cubes, a fourth power or in general any power whatever into two powers of the same denomination above the second is impossible, and I have assuredly found an admirable proof of this, but the margin is too narrow to contain it.” Mathematics professor Andrew Wiles proved the theorem in 1993.

Pierre de Fermat, who is best remembered for what is known as Fermat's Last Theorem (or FLT among the mathematicians), which states that xn+ yn= z n has no non-zero integer solutions for x, y and z when n > 2. Fermat wrote, in the margin of Bachet's translation of Diophantus's Arithmetica, "I have discovered a truly remarkable proof which this margin is too small to contain".

The truth of Fermat's assertion was proved in June 1993 by the British mathematician Andrew Wiles, but Wiles withdrew the claim to have a proof when problems emerged later in 1993. In November 1994 Wiles again claimed to have a correct proof, which has now been accepted. Unsuccessful attempts to prove the theorem over a 300 year period led to the discovery of commutative ring theory and a wealth of other mathematical discoveries.

I could attach the proof here for a smartarse bonus but the mail service provider here has too small a capacity.

Pierre de Fermat (1601-1665) made the marginal note (that x to the nth power plus y to the nth power cannot equal z to the nth power if n does not equal 2) and Andrew Wiles finally "fulfilled the quest" in 1994 by proving it. Wiles had been assisted by Richard Taylor and had built upon the work of Yutaka Taniyama/Goro Shimura, Ken Ribet, Gerhard Frey, and Jean-Pierre Serre.

Karl Marx, Friedrich Engels, Vladimir Ilich Lenin, Joseph Stalin, Chairman Mao, and Sun Yatsen. Explanation: The portraits of the founding fathers of Marxism-Leninism have been displayed on Tiananmen Square on every occasion of national importance: Marx and Engels on the side of the Museum of the Chinese Revolution, Lenin and Stalin on the side of the Great Hall of the People, all facing Mao's portrait affixed to the Forbidden City. The portrait of Sun Yatsen, the father of the Chinese Revolution, is usually placed in the middle. Irreverent Chinese and some western journalists call these portraits "The History of Shaving": from the bearded Marx and Engels to the goateed Lenin to the mustachioed Sun and Stalin, culminating in the hairless face of Mao.

The furniture belonged to Joel (Tom Cruise) -- or more accurately to his parents, played by Nicholas Pryor and Janet Caroll -- in Risky Business. "I" am Joe Pantoliano, and the other references are to The Matrix and Memento.

The pictures were, Johannes Brahms, an elephant's trunk, a (seemingly dead, but at the very least upended) rat, and Franz Liszt. The common link is that they all represent slang terms for being drunk, as follows: Brahms and Lizst, pissed -- from Cockney rhyming slang elephants trunk, drunk (or just elephants) -- also from Cockney rhyming rat-arsed (or just ratted) The amphibian that fits the pattern is the newt, as in: pissed as a newt.

Martin has obviously researched this question thoroughly.

State of utter inebriation. Brahms and Liszt

Elephants Trunk

Rat Arsed

Pissed as a Newt.

The lovers are #18 and #10.

Explanation: The clues refer to the International Radio Operators Alphabet, as follows: Alpha, Bravo, Charlie, Delta, Echo, Foxtrot, Golf, Hotel, India, Juliet, Kilo, Lima, Mike, November, Oscar, Papa, Quebec, Romeo, Sierra, Tango, Uniform, Victor, Whiskey, X-Ray, Yankee, Zulu. #6 is Foxtrot, #20 is Tango. #12 is Lima (capital city of Peru). Clearly, the lovers are Romeo (#18) and Juliet (#10). Though "Papa" and "Whiskey" might pull a close second...

Actors who have played Starfleet Captains.

aehnrst .........William Shatner -Captain James T. Kirk on Star Trek

aersttw .........Patrick Stewart - Captain Jean-Luc Picard Star Trek: The Next Generation

bkoors ..........Avery Brooks - Captain Benjamin Sisko, the Starfleet Captain in charge of the Deep Space Nine

eglmruw .......Kate Mulgrew - Captain Kathryn Janeway, the captain of the starship U.S.S. Voyager

aabklu .......Scott Bakula - Captain Jonathan Archer in UPN's Enterprise.

x786 -- to complete "cogito ergo sum".

39 degrees 45' (Denver CO). The others are the latitudes of state capitals alphabetically by state: Montgomery AL, Juneau AK, Pheonix AZ, Little Rock AR, and Sacramento CA

Between pragmatism and presentiment you find prank, prattle, prawn, precipice, pregnant, prejudicial, preoccupation and presbytery. You also find predestination.

The National flag of Kalmykia, a Republic within the Russian Federation. The Kalmyk people are the last Western Mongols, and the only Buddhist people in Europe.