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October 5th, 2010, 05:51 AM   #1
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Problems driving me crazy.(Primes,Fermat,Euler)

1) Show that if a is an integer and mdc(a,9)=1, so a^7 congruent a (mod 63).
Comments:
?(7) = 6 e ?(9) = 6 helps in any way?



2)Show that if n is a pseudoprime for the base a but not a pseudoprime for the base b, so n is not a pseudoprime for the base ab.
Comments:
Should I use a^b-1 congruent 1 (mod n) or a^n congruent a (mod n) ?



3) Prove that if ?(n) is prime, then n=3,4 or 6.
Comments:
Computing ?(3), ?(4) e ?(6) gives us prime numbers, but how to give a normal proof?
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October 5th, 2010, 09:30 AM   #2
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Re: Problems driving me crazy.(Primes,Fermat,Euler)

Quote:
Originally Posted by usermind
1) Show that if a is an integer and mdc(a,9)=1, so a^7 congruent a (mod 63).
Comments:
?(7) = 6 e ?(9) = 6 helps in any way?
mdc?

Generally, you'll want to work mod 7 and mod 9 then crt the results together.

Quote:
Originally Posted by usermind
2)Show that if n is a pseudoprime for the base a but not a pseudoprime for the base b, so n is not a pseudoprime for the base ab.
Comments:
Should I use a^b-1 congruent 1 (mod n) or a^n congruent a (mod n) ?
The former, then the identity property.

Quote:
Originally Posted by usermind
3) Prove that if ?(n) is prime, then n=3,4 or 6.
Comments:
Computing ?(3), ?(4) e ?(6) gives us prime numbers, but how to give a normal proof?
That proves that if n = 3, 4, or 6 then ?(n) is prime, but you were asked the converse.
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