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1. Show that a 1728 = 1 (mod p) when p= 7, 13, 19 for all a E N such that p /a. 2. Let p be a prime and p = 3 (mod 4). Show t

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1. P=7 éxa gcd caip) = 1 >> ged ca 7) - ) By fermats Theorem, 7-1 = 1 (mod 7) a 298 -) 96-1 (mod 7) -> (26) 1288 (mod 7) (mo2 19-1 a 1 (mod 19) als imod 19) ( 918 ) 96 = 196 (mod 19) =) 91728 El (mod 19) 9 1928 & 1 (mod P) when p=7,1319 P- / (mod p)3 » P-3 = 4k - P = uk + 3 Uk +3-) TE(-1) 2 (mod p) 4672 (mod P) a) 1 (-1) (2k+1) ? - 19 (-1) 2 (mod P) 2k +1 (mod P) - 1 (-1)

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