A:
g(k) = "For every
k, we denote by
g(
k) the minimum number
s of
k[sup]th[/sup] powers needed to represent all integers. Note we have
g(1) = 1. Some simple computations show that 7 requires 4 squares, 23 requires 9 cubes, and 79 requires 19 fourth-powers; these examples show that
g(2) ≥ 4,
g(3) ≥ 9, and
g(4) ≥ 19. Waring conjectured that these values were in fact the best possible."
G(k) = "From the work of
Hardy and
Littlewood, more fundamental than
g(
k) turned out to be
G(
k), which is defined to be the least positive integer
s such that every
sufficiently large integer (i.e. every integer greater than some constant) can be represented as a sum of at most
s k[sup]th[/sup] powers of positive integers. Since squares are congruent to 0, 1, or 4 (mod 8), no integer congruent to 7 (mod 8) can be represented as a sum of three squares, implying that
G(2) ≥ 4. Since
G(
k) ≤
g(
k) for all
k, this shows that
G(2) = 4.
Davenport showed that
G(4) = 16 in 1939, by demonstrating that any sufficiently large number congruent to 1 through 14 mod 16 could be written as a sum of 14 fourth powers (Vaughan in 1985 and 1989 reduced the 14 successively to 13 and 12). The exact value of
G(
k) is unknown for any other
k, but there exist bounds."
Q: Why is battlefield 3 the titties, and why does back to karkand kick so much ass that it makes CoD look like poop?
bonus Q: why do so many losers keep asking so many obscure/technical questions...