1)/2) = 3, so this method generates three solutions to each Pell equation, as follows. For q = 1, the first three solutions to the Pell equation x2 − 2y2 = 1 are (3,2), (17,12), and (99,70). Thus, for each of the three values xi = 3, 17, and 99, Lehmer's method tests the pair (xi − 1)/2, (xi + 1)/2 for smoothness; the three pairs to be tested are (1,2), (8,9), and (49,50). Both (1,2) and (8,9) are pairs of consecutive P-smooth numbers, but (49,50) is not, as 49 has 7 as a prime factor. For q = 3, the first three solutions to the Pell equation x2 − 6y2 = 1 are (5,2), (49,20), and (485,198). From the three values xi = 5, 49, and 485 Lehmer's method forms the three candidate pairs of consecutive numbers (xi − 1)/2, (xi + 1)/2: (2,3), (24,25), and (242,243). Of these, (2,3) and (24,25) are pairs of consecutive P-smooth numbers but (242,243) is not. For q = 5, the first three solutions to the Pell equation x2 − 10y2 = 1 are (19,6), (721,228), and (27379,8658). The Pell solution (19,6) leads to the pair of consecutive P-smooth numbers (9,10); the other two solutions to the Pell equation do not lead to P-smooth pairs. For q = 6, the first three solutions to the Pell equation x2 − 12y2 = 1 are (7,2), (97,28), and (1351,390). The Pell solution (7,2) leads to the pair of consecutive P-smooth numbers (3,4). For q = 10, the first three solutions to the Pell equation x2 − 20y2 = 1 are (9,2), (161,36), and (2889,646). The Pell solution (9,2) leads to the pair of consecutive P-smooth numbers (4,5) and the Pell solution (161,36) leads to the pair of consecutive P-smooth numbers (80,81). For q = 15, the first three solutions to the Pell equation x2 − 30y2 = 1 are (11,2), (241,44), and (5291,966). The Pell solution (11,2) leads to the pair of consecutive P-smooth numbers (5,6). For q = 30, the first three solutions to the Pell equation x2 − 60y2 = 1 are (31,4), (1921,248), and (119071,15372). The Pell solution (31,4) leads to the pair of consecutive P-smooth numbers (15,16). Counting solutions Størmer's original result can be used to show that the number of consecutive pairs of integers that are smooth with respect to a set of k primes is at most 3k −