1,105 belongs to a small, strange family of numbers that manage to fool a genuinely reasonable primality test into calling them prime when they aren't. That family is called Carmichael numbers, named after the American mathematician Robert Carmichael, who studied them in the early 20th century, and 1,105 is the second one, following 561.
Fooling the Fermat Test
The test they fool is built on Fermat's Little Theorem, which states that if p is prime and a is any number not divisible by p, then a^(p−1) leaves a remainder of exactly 1 when divided by p. This gives a genuinely useful, fast way to screen for primality: pick a base a, compute a^(n−1) mod n, and if the result isn't 1, n is definitely composite. The catch is that passing this test for one or even several bases doesn't guarantee primality — and Carmichael numbers are exactly the composite numbers that pass this specific test for *every* valid base simultaneously, making them a genuine, structural blind spot for this particular shortcut rather than a rare fluke tied to one unlucky choice of base.
Checking the Korselt Condition
1,105 factors as 5 × 13 × 17, and its Carmichael status follows from a specific structural condition (established by the mathematician Alwin Korselt in 1899, actually predating Carmichael's own published work by over a decade): a composite, square-free number n is a Carmichael number exactly when, for every prime p dividing n, (p − 1) evenly divides (n − 1). Checking this for 1,105: n − 1 = 1,104. Does (5−1)=4 divide 1,104 evenly? 1,104 ÷ 4 = 276, yes. Does (13−1)=12 divide 1,104? 1,104 ÷ 12 = 92, yes. Does (17−1)=16 divide 1,104? 1,104 ÷ 16 = 69, yes. All three conditions hold, confirming 1,105's Carmichael status through direct, checkable arithmetic rather than an appeal to authority.
Because trial-division primality checks aren't practical for the enormous numbers used in real cryptographic systems, modern software instead relies on fast probabilistic tests built on exactly this same Fermat-style logic — which is precisely why Carmichael numbers matter well beyond pure recreational curiosity: any cryptographic system relying on a naive Fermat test alone would be vulnerable to being fooled by a large enough Carmichael number, which is part of why real-world implementations use more sophisticated tests (like the Miller-Rabin test) specifically designed to catch what plain Fermat testing misses.
Four Ways to Sum Two Squares
1,105 carries an entirely separate, genuinely elegant mathematical property worth knowing alongside its Carmichael status: it can be written as the sum of two perfect squares in four different ways — 1,105 = 4² + 33² = 9² + 32² = 12² + 31² = 23² + 24². This unusually rich set of representations isn't a coincidence; it follows directly from 1,105 being a product of three distinct prime factors (5, 13, and 17) that are each themselves expressible as a sum of two squares and each individually congruent to 1 modulo 4 — a genuine structural property from number theory (related to a classical theorem first proven rigorously by Fermat) rather than an accident specific to this one number.
The first Carmichael number, 561 (= 3×11×17), is worth knowing alongside 1,105 as a point of comparison — it was actually identified even earlier, in 1910, by the mathematician Alwin Korselt, though he didn't fully recognize its significance at the time, and it was Carmichael's independent 1910 work naming and studying the broader category that the family eventually took its name from. Carmichael numbers become progressively rarer, though not nearly as dramatically as perfect numbers do, as numbers grow larger — they were eventually proven, in 1994, to exist in infinite supply (a genuinely significant proof, resolving a question that had remained open for over eighty years since Carmichael's original work).
That 1994 proof, by mathematicians Red Alford, Andrew Granville, and Carl Pomerance, resolved a question that had sat open since Carmichael's own era, and it's a useful data point for appreciating how differently "genuinely open" and "eventually resolved" problems can play out even within the same broad neighborhood of number theory: the Carmichael-number infinitude question took roughly eight decades to settle, while the parallel question of whether odd perfect numbers exist at all, discussed on several of this site's other perfect-number pages, has remained open for well over two thousand years with no comparable resolution in sight.