Consider the open sentenceP(m):5m + 1 = a2 for some a Z,where m N. That is, P(m) is the

Chapter 4, Problem 4.98

(choose chapter or problem)

Consider the open sentenceP(m):5m + 1 = a2 for some a Z,where m N. That is, P(m) is the open sentence: 5m + 1 is a perfect square.(a) Determine four distinct solutions t of t 2 4 (mod 5). For each solution t, determine m = 4(t 24)5 + 3and show that P(m) is a true statement.(b) Show that the set S = {t Z : t 2 4 (mod 5)} contains infinitely many elements.(c) Let t be an element of the set S in (b). Prove that if m = 4(t 24)5 + 3, then 5m + 1 is a perfect square.(d) As a consequence of the results established in (a)(c), what can be concluded about the setM = {m N : 5m + 1 is a perfect square}?