Solution
Using the sum of cubes factorisation, $$3^{21}+1 = (3^7+1)(3^{14} - 3^7 + 1) = 2188 \times 4780783$$. Here $$2188 = 2^2 \times 547$$ and $$4780783 = 7^2 \times 43 \times 2269$$. The largest prime factor is therefore 2269.
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