By Fosner A., Fosner M.

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We know that if 2304167 is not prime then it has a prime divisor 2304617 1518. After 14 more trial divisions, we would nd that the prime 19 58 + 1 = 1103 divides 2304167. p Dividing, we have 2304167=1103 = 2089. If 2089 were not prime, then it would have a prime factor 2089 46, but also 1 mod 58. There aren't any such things, so 2089 is prime. 4 Factoring 3n - 1 We continue with more examples using Fermat's observation about factors of special numbers of the form bn , 1. Every number 3n , 1 for n 1 has the obvious factor 3 , 1, so is not prime.

Thus, 12 are square roots of -1 modulo 29. Similarly, we nd that 5 are square roots of -1 modulo 13, and 4 are square roots of -1 modulo 17. To use Sun Ze's theorem to get a solution modulo 6409 = 13 17 29 from this, we rst need integers s; t so that s 13 + t 17 = 1. square roots of -1 modulo 13 17 29, since we already have the numbers s,t" in our possession. 4 Hensel's Lemma for prime-power moduli In many cases, solving a polynomial equation f x 0 mod p modulo a prime p su ces to assure that there are solutions modulo pn for powers pn of p, and also to nd such solutions e ciently.

The argument is by counting: we'll count the number of numbers x in the range from 0 through N , 1 which do have a common factor with N , and subtract. And, by unique factorization, if x has a common factor with N then it has a common prime factor with N . There are exactly N=pi numbers divisible by pi , so we would be tempted to say that the number of numbers in that range with no common factor with N would be 1 N , pN , pN , : : : pN 1 2 n However, this is not correct in general: we have accounted for numbers divisible by two di erent pi 's twice, so we should add back in all the expressions N=pi pj with i 6= j .

### 2-local superderivations on a superalgebra Mn(C) by Fosner A., Fosner M.

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