Pell's Equation is a very simple Diophantine equation that has been known to mathematicians for over 2000 years. Even today research involving this equation continues to be very active, as can be seen by the publication of at least 150 articles related to this equation over the past decade. However, very few modern books have been published on Pell's Equation, and this will be the first to give a historical development of the equation, as well as to develop the necessary tools for solving the equation.

The authors provide a friendly introduction for advanced undergraduates to the delights of algebraic number theory via Pell's Equation. The only prerequisites are a basic knowledge of elementary number theory and abstract algebra. There are also numerous references and notes for those who wish to follow up on various topics.

Rivers were the arteries of our first civilizations - the Tigris and Euphrates of Mesopotamia, India's Ganges, Egypt's Nile, the Yellow River of China - and have nourished modern cities from London to New York, so it is natural that poets have for centuries drawn essential meanings and metaphors from their endless currents. English poets from Shakespeare and Dryden, Wordsworth and Byron to Ted Hughes, John Betjeman and Alice Oswald; Irish poets - Eavan Boland, Seamus Heaney, Paul Muldoon, to name but a few; Scottish and Welsh poets from Henry Vaughan and Robert Louis Stevenson to Robin Robertson and Gillian Clarke. A whole raft of American poets from Whitman, Emerson and Emily Dickinson to Langston Hughes, Mary Oliver, Natasha Trethewey and Grace Paley. Folk songs. African-American spirituals. Poems from ancient Egypt and Rome. From medieval China and Japan. And a truly international selection of modern poets from Europe (France, Italy, Russia, Serbia), India, Africa, Australia and South and Central America, all combining in celebration of the rivers of the world. From the Mississippi to the Limpopo. From the Dart to the Danube. Plunge in.

What are the 50 key events you need to understand to grasp British history?

If you could choose the 50 things that define British history, events of significance not only in themselves, but in their importance to wider themes running through our past, what would they be? Hugh Williams has made that selection, and the result is a fascinating overview of Britain s past.

He refines British history into a series of key themes that represent a crucial strand in our history, and pinpoints the seminal events within those strands - Roots, from the Roman invasion to Britain s entry into the Common Market; Fight, Fight and Fight Again, from the Battle of Agincourt to the Falklands War; The Pursuit of Liberty, from the Magna Carta through the Glorious Revolution to the foundation of the NHS; Home and Abroad, from Sir Francis Drake and Clive of India to the arrival of the SS Empire Windrush; and All Change, from Chaucer and the English language to the invention of the jet engine.

With great clarity, simplicity and a zest for the marvellous stories that underpin many of these events, Hugh Williams explains the linkage between each one and its importance in the progress of British history as a whole. Along the way, he has some fascinating tales to tell, making this a highly enjoyable read as well as a perceptive insight into our shared past, and vital for anyone who wants quickly and enjoyably to grasp the essential facts about Britain s history."

Pell's Equation is a very simple Diophantine equation that has been known to mathematicians for over 2000 years. Even today research involving this equation continues to be very active, as can be seen by the publication of at least 150 articles related to this equation over the past decade. However, very few modern books have been published on Pell's Equation, and this will be the first to give a historical development of the equation, as well as to develop the necessary tools for solving the equation.

The authors provide a friendly introduction for advanced undergraduates to the delights of algebraic number theory via Pell's Equation. The only prerequisites are a basic knowledge of elementary number theory and abstract algebra. There are also numerous references and notes for those who wish to follow up on various topics.