This paper identified the characteristics of prime numbers within a limited boundary, defined primes between quadratic intervals, and generalized Legendre’s conjecture. Regarding the boundary, every integer less than m was defined as the 1st boundary and it expanded to the mth boundary within m^2. Thus, each boundary contained m elements. Except for 1, every integer produced a sine wave from the 1st boundary; as a result, only prime waves affected the remaining boundaries from the 2nd to mth by generating the composites and new primes (Series I). Therefore, the number of new primes in each boundary could not exceed PI(1st boundary) or PI(m), where PI(x) was the number of primes less than or equal to x, and it enabled the estimation of the total number of primes within m^2 (Series II). Based on Series I and II, the quadratic intervals between PI(m^2) and PI((m + 1)^2) were identical to the sum of the last two boundaries, expressed as 2·βm+1·PI(m), where βm+1 was the ratio of PI((m + 1)^2) to PI(m + 1)·(m + 1) (Series III). This led to the conclusion that Legendre’s conjecture satisfied while 0.8986·PI(P) < 2·βP·PI(P) < PI(P) (prime P > 113), or 2·βm+1·PI(m) ≤ 2·PI(m) (integer m ≥ 2).