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MA2317: Introduction to Number Theory Tutorial problems, November 27, 2010 √ √ 1. Show that 2 + P3 is an algebraic number. 1 2. Show that e = is irrational. (Hint: e can be approximated n! n>0 P 1 = satisfies by rational numbers too well: the rational number qpm n! m 06n6m | qpm m 1 .) mqm − e| < 3. Show that the polynomial xn +2(xn−1 +xn−2 +. . .+x)+4 is irreducible over integers for n > 3. 4. Find a rational parametrisation of the curve y2 + 2x2 − x − y − 1 = 0.