The smallest integers that can be written as a sum of two cubes in three different ways, then four different ways, and so on, are often called taxicab numbers.
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Although the four operations—addition, subtraction, multiplication and division—are studied from elementary school onward, students must wait until their third year of secondary school to discover a fifth: raising a number to a power.
Since the 18th century, a famous problem—Waring's conjecture—has challenged mathematicians. Every integer N is a sum of at most four squares, or of at most nine cubes. Likewise, given an integer n, can every N be expressed as a sum of at most g(n) nth powers, and if so, what is the smallest possible value of g(n)?
With the development of probability theory in the early 20th century, a new field of inquiry opened up in arithmetic: the statistical study of integers. Paul Erdős was among the first mathematicians to grasp the significance of this new approach.
The most widespread cryptography system relies on the use of very large integers, whose factorization remains beyond the reach of our computers. As early as the 17th century, Mersenne and Fermat were investigating the prime factorization of very large numbers; their work inspired modern factorization algorithms.
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