[seqfan] Primes and Giuga numbers in relation to Bernoulli numbers

[Tomasz Ordowski]

Dear readers!

Let B(k) = N(k) / D(k) be the k-th Bernoulli number

Conjecture: a number p > 1 is prime if and only if
   p N(p-1) == - D(p-1) (mod p^2).
Is this equivalent to the Agoh-Giuga conjecture?
See https://en.wikipedia.org/wiki/Agoh%E2%80%93Giuga_conjecture

Conjecture: a composite m is a Giuga number if and only if
   m N(phi(n)) == - D(phi(m)) (mod m^2).
Cf. https://en.wikipedia.org/wiki/Giuga_number (see the definition by
Agoh).
Note: Euler's phi function can be replaced by the Carmichael lambda
function.

Let a(n) = Numerator(N(phi(n))/n + D(phi(n))/n^2),
and a'(n) = Numerator(N(lambda(n))/n + D(lambda(n))/n^2).
It should be noted that a(n) = Numerator(B(phi(n)) + 1/n)
and a'(n) = Numerator(B(lambda(n)) + 1/n).

Let b(n) = Denominator(N(phi(n))/n + D(phi(n))/n^2).
Note that b(n) = Denominator(N(lambda(n))/n + D(lambda(n))/n^2).
Conjecture: for n > 1, b(n) = 1 if and only if n is a prime or a Giuga
number.

Best regards,

Thomas Ordowski
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https://oeis.org/A309132