A composite number that passes the Baillie–PSW test

Find an odd composite number passing both the strong base-2 Fermat test and the strong Lucas test of Baillie–PSW; none is known below 2^64.

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1	# A composite number that passes the Baillie–PSW test
2
3	The Baillie–PSW test calls an odd number "probably prime" when it passes a strong Fermat test to base 2 and a strong Lucas test. It is fast and is used in computer algebra systems such as Maple and SageMath, yet in more than forty years nobody has found a composite number that passes it. The task is to find one.
4
5	## Problem
6
7	Let $n > 1$ be odd.
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9	- Strong Fermat test to base 2. Write $n - 1 = d \cdot 2^s$ with $d$ odd. The number $n$ passes if $2^d \equiv 1 \pmod n$, or $2^{d \cdot 2^r} \equiv -1 \pmod n$ for some $0 \le r < s$.
10	- Strong Lucas test (Selfridge's method A). Let $D$ be the first term of $5, -7, 9, -11, 13, \dots$ whose Jacobi symbol $\left(\frac{D}{n}\right)$ equals $-1$ (no such $D$ exists when $n$ is a perfect square, so squares are excluded first). Put $P = 1$ and $Q = (1 - D)/4$, and define the Lucas sequences $U_0 = 0$, $U_1 = 1$, $V_0 = 2$, $V_1 = P$, with $W_k = P W_{k-1} - Q W_{k-2}$ for $W \in \{U, V\}$. Write $n + 1 = e \cdot 2^t$ with $e$ odd. The number $n$ passes if $U_e \equiv 0 \pmod n$, or $V_{e \cdot 2^r} \equiv 0 \pmod n$ for some $0 \le r < t$.
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12	Task. Find an odd composite $n$ that passes both tests (a Baillie–PSW pseudoprime), or prove that none exists.
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14	## Background
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16	The combined test first appeared in 1980 in a paper of Pomerance, Selfridge and Wagstaff, whose exhaustive search found no composite number below $25 \cdot 10^9$ passing both parts; the idea of combining the two tests originated with Baillie [2, 3], and Baillie and Wagstaff analysed the Lucas side [1]. Each test on its own is passed by some composites (OEIS [A001262](https://oeis.org/A001262) and [A217255](https://oeis.org/A217255)); the hope behind the combination is that the two kinds of failure rarely meet. In a 1984 note Pomerance argued heuristically that they do meet: the number of counterexamples up to $x$ should exceed $x^{1-\varepsilon}$ for every $\varepsilon > 0$ [3]. So counterexamples are expected to exist, but they have never been seen.
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18	## What is known
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20	- No Baillie–PSW pseudoprime exists below $2^{64}$, by exhaustive computation (as stated by Epoch).
21	- Pomerance's heuristic predicts more than $x^{1-\varepsilon}$ counterexamples up to $x$ [3].
22	- Baillie, Fiori and Wagstaff (2021) strengthened the test at almost no extra cost by adding a check on what they call Lucas-V pseudoprimes, of which there are only five below $10^{15}$ [4].
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24	## What would count as progress
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26	- An odd composite $n$ together with a factorisation $n = ab$, $a, b > 1$, such that $n$ passes both tests. Checking takes milliseconds with arbitrary-precision arithmetic.
27	- Epoch's verifier takes two integers greater than 1 and runs both tests exactly on their product, with no size limit.
28	- A method that produces many such numbers would be stronger still; a proof that none exist would settle the question the other way.
29
30	## References
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32	1. R. Baillie and S. S. Wagstaff Jr., Lucas pseudoprimes, Math. Comp. 35 (1980), 1391–1417. [doi:10.1090/S0025-5718-1980-0583518-6](https://doi.org/10.1090/S0025-5718-1980-0583518-6)
33	2. C. Pomerance, J. L. Selfridge and S. S. Wagstaff Jr., The pseudoprimes to $25 \cdot 10^9$, Math. Comp. 35 (1980), 1003–1026. [doi:10.1090/S0025-5718-1980-0572872-7](https://doi.org/10.1090/S0025-5718-1980-0572872-7)
34	3. C. Pomerance, Are there counter-examples to the Baillie–PSW primality test?, 1984. [math.dartmouth.edu/~carlp/dopo.pdf](https://math.dartmouth.edu/~carlp/dopo.pdf)
35	4. R. Baillie, A. Fiori and S. S. Wagstaff Jr., Strengthening the Baillie-PSW primality test, Math. Comp. 90 (2021), 1931–1955. [doi:10.1090/mcom/3616](https://doi.org/10.1090/mcom/3616)
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37	## Source
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39	Imported from [FrontierMath: Open Problems, baillie-psw](https://epoch.ai/frontiermath/open-problems/baillie-psw), curated by Epoch AI. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.