The K^th "Schur number" S(K) is the least positive integer N such that for every coloring of the integers {1,2,3,...,N} with K colors, an equation a+b=c exists with 1≤a≤b≤c≤N with a,b,c all having the same color. Equivalently it is the greatest possible N so that a (K-1)-coloring of {1,2,3,...,N-1} exists avoiding monochromatic a+b=c. [The latter formulation has the advantage of telling us that S(0)=1.] We give very simple proofs that G^K≤S(K)≤⌈K!(e-1/24)⌉ where e≈2.71828 for various growth-factors G=3/2, G=2, and (for k≥5) G=161^1/5=2.76290... These Gs are not the best known, but my proofs are very simple. We speculate that S(K) grows superexponentially, and (if that is true) explain how a computer could prove arbitrarily large Gs or (if it is not true) prove a G arbitrarily near to maximum possible, after enough work. I have not been able to prove superexponentiality, but the final paragraph explains a strategy that might be able to.