pell

Pell is a surname that appears in several mathematical concepts and in U.S. higher education funding. In mathematics, the term is most often associated with Pell's equation, the Pell numbers, and Pell's identity. The naming traces to the English mathematician John Pell, though the historical attribution was later clarified by scholars.

Pell's equation refers to the Diophantine equation x^2 − D y^2 = 1, where D is a non-square integer. For such D, there are infinitely many integer solutions (x, y). Solutions can be generated from the continued fraction expansion of sqrt(D). The equation plays a central role in algebraic number theory and the study of units in quadratic fields.

The Pell numbers form a sequence defined by P_0 = 0, P_1 = 1, and P_n = 2 P_{n-1} + P_{n-2} for n ≥ 2. They grow roughly like (1 + √2)^n and have a closed-form expression P_n = ((1 + √2)^n − (1 − √2)^n) / (2√2). The sequence is related to approximations of √2 and to certain quadratic recurrences.

Pell's identity expresses a multiplicative composition for expressions of the form x^2 − D y^2. A common form is (x^2 − D y^2)(u^2 − D v^2) = (xu + D yv)^2 − D (xv + yu)^2, which helps generate new solutions from known ones.

Pell Grant is a U.S. federal undergraduate need-based financial aid program established by the Higher Education Act of 1965. It provides funds that do not require repayment, with awards determined by demonstrated financial need and available appropriations. The name has remained in common usage as a shorthand for this program.