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Walker 1976 consecutive integer pairs powerful numbers related
example_p116: Walker's example that the odd powers of the seventh power of 2 sqrt(7) + 3 sqrt(3), the smallest solution of 7 X^2 - 3 Y^2 = 1, give infinitely many consecutive powerful pairs with neither member a square, the first being 48,689,748,233,307 and 48,689,748,233,308.
theorem_2_2: Walker's criterion for property Q, every prime of D dividing v, among the solutions u + v sqrt(D) of X^2 - D Y^2 = +-1: none with the minus sign and D even, all if the fundamental solution has it, and otherwise the least such solution is the i-th power of the fundamental solution x + y sqrt(D), with i the product of the distinct odd primes dividing D but not y.
theorem_2_5: Walker's theorem that the powers of the least solution with property Q of X^2 - D Y^2 = +-1 give all positive solutions with property Q, every power for the plus sign and the odd powers for the minus sign, so that they account for all consecutive powerful pairs with a square member.
theorem_3_2: Walker's criterion for property Q, every prime of mn dividing uv, among the solutions u sqrt(m) + v sqrt(n) of m X^2 - n Y^2 = +-1: none if m (or n) is even and x (or y) odd, all if the smallest solution has it, and otherwise the least such solution is the odd power 2i + 1 of the smallest solution, with 2i + 1 the product of the distinct odd primes dividing mn but not xy.
theorem_3_5: Walker's theorem that the odd powers of the least solution with property Q of m X^2 - n Y^2 = +-1 give all its positive solutions with property Q, so that they account for all consecutive powerful pairs with neither member a square.
Walker, David T., Consecutive integer pairs of powerful numbers and related Diophantine equations. Fibonacci Quart. 14(2) (1976), 111-116.
Following Golomb's split of consecutive powerful-number pairs into Type I (one member a perfect square) and Type II (neither a square), Walker gives a complete description of both. Type I pairs correspond to solutions of the Pell equation X^2 - D Y^2 = +-1 in which D Y^2 is powerful, which he encodes as 'property Q': a solution u + v sqrt(D) has property Q if every prime dividing D also divides v. Theorem 2.1 recalls how powers of the fundamental solution generate all positive solutions, and Theorem 2.2 settles when property Q occurs: with the minus sign and D even no solution has it; if the fundamental solution has property Q then all positive solutions do; otherwise the least solution with property Q, when it exists, is the i-th power of the fundamental solution x + y sqrt(D), with i the product of those odd primes that divide D and do not divide y. Lemmas 2.3 and 2.4 show property Q is preserved under multiplication of solutions and relate the least such solutions for the two signs, and Theorem 2.5 shows the powers of the least property-Q solution (odd powers for the minus sign) give all positive property-Q solutions. The second part of the paper treats Type II pairs through solutions of m X^2 - n Y^2 = +-1 (Theorems 3.2 and 3.5), and its closing example (p. 116) is a Type II pair, 48,689,748,233,307 = 3(4,028,637)^2 and 48,689,748,233,308 = 7(2,637,362)^2, neither a square, whose defining solution of 7X^2 - 3Y^2 = 1 has odd powers giving infinitely many more. The paper concludes that the two parts together account for all pairs of consecutive powerful numbers. It gives no count of the pairs up to x.
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Bears on. #365: the example on p. 116, with Theorems 3.2 and 3.5, gives infinitely many pairs of consecutive powerful numbers with neither member a square, so the first question, whether one member must be a square, has answer no; Theorems 2.2, 2.5, 3.2 and 3.5 describe all pairs, with and without a square member, through solutions of X^2 - D Y^2 = +-1 and m X^2 - n Y^2 = +-1. The paper gives no count of the pairs up to x, which the second question asks about.
Results. Theorem 2.2 (p. 112); Theorem 2.5 (p. 113); Theorem 3.2 (p. 115); Theorem 3.5 (p. 115); the example of 7X^2 - 3Y^2 = 1 (p. 116, unnumbered). Theorems 2.1 (p. 111) and 3.1 (p. 114) are recalled without proof and are stated on the pages of Theorems 2.2 and 3.2 where used; Lemmas 2.3 and 2.4 (pp. 112-113) and 3.3 and 3.4 (p. 115) are proof steps, summarized on the pages of Theorems 2.5 and 3.5.
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