<jupyter_start><jupyter_text>Euler Problem 120Let $r$ be the remainder when $(a−1)^n + (a+1)^n$ is divided by $a^2$.For example, if $a = 7$ and $n = 3$, then $r = 42$: $6^3 + 8^3 = 728 \equiv 42 \pmod{49}$.And as $n$ varies, so too will $r$, but for $a = 7$ it turns out that $r_{max} = 42$.For $3 \le a \le 1000$, find $\sum r_{max}$.<jupyter_code>def rmax(a):
    t1 = t2 = 1
    aa = a*a
    Table = {(1,1):1}
    rm = 2 % aa
    while 1:
        t1 = ((a-1)*t1) % aa
        t2 = ((a+1)*t2) % aa
        rm = max(rm, (t1+t2)%aa)
        if (t1, t2) in Table:
            return rm
        Table[(t1,t2)] = 1

print(sum(rmax(a) for a in range(3,1001)))<jupyter_output>333082500
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