kill(all)$

van=facebook.com/LearningCalculus$

f(a,b,c,d,e):=a^1+b^1+c^1+d^1=e$
g(a,b,c,d,e):=a^2+b^2+c^2+d^2=e$
h(a,b,c,d,e):=a^3+b^3+c^3+d^3=e$

max: 106/2$

for a:0 thru max step 1 do
    for b:a thru max step 1 do
        for c:b thru max step 1 do
            for d:c thru max step 1 do
            if f(a,b,c,d,106) then
                if g(a,b,c,d,4134) then
                    if h(a,b,c,d,179776) then
                    (
                        print("Answer: ", a, b, c, d)
                    )$

max: 274/2$

for a:0 thru max step 1 do
    for b:a thru max step 1 do
        for c:b thru max step 1 do
            for d:c thru max step 1 do
            if f(a,b,c,d,274) then
                if g(a,b,c,d,27674) then
                    if h(a,b,c,d,3115654) then
                    (
                        print("Answer: ", a, b, c, d)
                    )$