Four squares of primes and 165 powers of 2

Four prime squares and powers of 2

“…In view of these results and Lagrange's theorem of four squares, it is reasonable to conjecture that every large even integer n ≡ 4 (mod 24) is a sum of four squares of primes n = p In [4], Liu and Liu showed that k = 8330 is acceptable in (1.2). Recently, Liu and Lü [7] improved this result and proved that k = 165 suffices.…”

“…This is Proposition 4.3 in [6] except for the value of c 3 . It has been shown in [6] that By the proof of Lemma 5.2 in [7] we have…”

“…Following the argument of [7], suppose first N ≡ 4 (mod 8), let E λ be defined in (2.8), and M and m as in (2.5) with P, Q determined in (2.1). Then (2.2) becomes…”

“…By Proposition 3 in [2], we know that the conclusion of Lemma 3.1 of [7] holds for D = N 1/16−2ε . By the argument in Section 3 of [7], in the proof of Lemma 2.2 of [7], we can fix z = N 1/32−ε , and then we can get c 1 ≤ (1+ε) 6 ·101·32 4 in Lemma 2.2 of [7]. Following the argument of the proof of Lemma 4.1 of [7], by Lemma 4 we get the assertion of Lemma 3.…”

On pairs of equations in unlike powers of primes and powers of 2

“…Later Liu and Lü [7] improved the value of k of (1.2) to 165, Li [3] improved it to 151 and Zhao [13] improved it to 46. Finally Platt and Trudgian [11] revised it to 45.…”