Smallest Strong Fibonacci Pseudoprime of Type I
Theorem
The smallest strong Fibonacci pseudoprime of type I is $443 \, 372 \, 888 \, 629 \, 441$.
Proof
Let $N := 443 \, 372 \, 888 \, 629 \, 441$, to save writing it out in full each time.
From the definition of a strong Fibonacci pseudoprime of type I:
A strong Fibonacci pseudoprime of type I is a Carmichael number $N = \ds \prod p_i$ such that an even number of the prime factors $p_i$ are of the form $4 m - 1$ where:
\(\text {(1)}: \quad\) | \(\ds 2 \paren {p_i + 1}\) | \(\divides\) | \(\ds \paren {N - 1}\) | for those $p_i$ of the form $4 m - 1$ | ||||||||||
\(\text {(2)}: \quad\) | \(\ds \paren {p_i + 1}\) | \(\divides\) | \(\ds \paren {N \pm 1}\) | for those $p_i$ of the form $4 m + 1$ |
We have that:
- $N = 17 \times 31 \times 41 \times 43 \times 89 \times 97 \times 167 \times 331$
Of these, we see that:
\(\ds 17\) | \(=\) | \(\ds 4 \times 4 + 1\) | which is a prime number of the form $4 n + 1$ | |||||||||||
\(\ds 31\) | \(=\) | \(\ds 4 \times 8 - 1\) | which is a prime number of the form $4 n - 1$ | |||||||||||
\(\ds 41\) | \(=\) | \(\ds 4 \times 10 + 1\) | which is a prime number of the form $4 n + 1$ | |||||||||||
\(\ds 43\) | \(=\) | \(\ds 4 \times 11 - 1\) | which is a prime number of the form $4 n - 1$ | |||||||||||
\(\ds 89\) | \(=\) | \(\ds 4 \times 22 + 1\) | which is a prime number of the form $4 n + 1$ | |||||||||||
\(\ds 97\) | \(=\) | \(\ds 4 \times 24 + 1\) | which is a prime number of the form $4 n + 1$ | |||||||||||
\(\ds 167\) | \(=\) | \(\ds 4 \times 42 - 1\) | which is a prime number of the form $4 n - 1$ | |||||||||||
\(\ds 331\) | \(=\) | \(\ds 4 \times 83 - 1\) | which is a prime number of the form $4 n - 1$ |
Thus there are $4$ (an even number) of prime factors of $N$ of the form $4 n - 1$.
We note that:
\(\ds N - 1\) | \(=\) | \(\ds 288 \times 1 \, 539 \, 489 \, 196 \, 630\) | and so $\paren {17^2 - 1} \divides \paren {N - 1}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 960 \times 461 \, 846 \, 758 \, 989\) | and so $\paren {31^2 - 1} \divides \paren {N - 1}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 1680 \times 263 \, 912 \, 433 \, 708\) | and so $\paren {41^2 - 1} \divides \paren {N - 1}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 1848 \times 239 \, 920 \, 394 \, 280\) | and so $\paren {43^2 - 1} \divides \paren {N - 1}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 7920 \times 55 \, 981 \, 425 \, 332\) | and so $\paren {89^2 - 1} \divides \paren {N - 1}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 9408 \times 47 \, 127 \, 220 \, 305\) | and so $\paren {97^2 - 1} \divides \paren {N - 1}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 27 \, 888 \times 15 \, 898 \, 339 \, 380\) | and so $\paren {167^2 - 1} \divides \paren {N - 1}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 109 \, 560 \times 4 \, 046 \, 850 \, 024\) | and so $\paren {331^2 - 1} \divides \paren {N - 1}$ |
Thus for all $p \divides N$, we have that $\paren {p^2 - 1} \divides \paren {N - 1}$.
From Difference of Two Squares we have that:
- $p^2 - 1 = \paren {p + 1} \paren {p - 1}$
and so for all $p \divides N$:
- $\paren {p - 1} \divides \paren {N - 1}$
We also have that $N$ is square-free.
Thus by Korselt's Theorem, $N$ is a Carmichael number.
Now we also have that:
\(\ds \paren {p^2 - 1}\) | \(\divides\) | \(\ds \paren {N - 1}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {p + 1} \paren {p - 1}\) | \(\divides\) | \(\ds \paren {N - 1}\) | Difference of Two Squares | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds 2 \paren {p + 1} \paren {\frac {p - 1} 2}\) | \(\divides\) | \(\ds \paren {N - 1}\) | as $p - 1$ is even | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds 2 \paren {p + 1}\) | \(\divides\) | \(\ds \paren {N - 1}\) |
Thus we have that:
- $\forall p \divides N: 2 \paren {p + 1} \divides \paren {N - 1}$
This is actually stronger than the conditions for which $N$ is a strong Fibonacci pseudoprime of type I:
A strong Fibonacci pseudoprime of type I is a Carmichael number $N = \ds \prod p_i$ such that an even number of the prime factors $p_i$ are of the form $4 m - 1$ where:
\(\text {(1)}: \quad\) | \(\ds 2 \paren {p_i + 1}\) | \(\divides\) | \(\ds \paren {N - 1}\) | for those $p_i$ of the form $4 m - 1$ | ||||||||||
\(\text {(2)}: \quad\) | \(\ds \paren {p_i + 1}\) | \(\divides\) | \(\ds \paren {N \pm 1}\) | for those $p_i$ of the form $4 m + 1$ |
It can be established by an exhaustive search that there are no smaller Carmichael numbers with this property.
Hence the result.
$\blacksquare$
Sources
- Jul. 1993: R.G.E. Pinch: The Carmichael Numbers up to $10^{15}$ (Math. Comp. Vol. 61, no. 203: pp. 381 – 391) www.jstor.org/stable/2152963
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $443,372,888,629,441$