The Hunt for Legendre Pairs of Length 153
Your computer can help settle a question mathematicians have been asking since 1893.A crowd-sourced search on DCP for two ±1 sequences of length 153 = 9 × 17 with the compression a conjecture predicts, whose autocorrelations sum to −2 at every shift. None is known. The same search confirmed the newly published pair of length 117 in October 2026, producing and verifying it in the two patches where it sits. Explore a region, sponsor the search, run a worker, and win from the prize pool if your patch holds a pair.
The space
Every patch is one slice of the search: one class of compressed pairs and a range of its lifts to length 153, laid along a Hilbert curve so the slices of a class stay neighbours. Black is unexplored, white is fully explored, red shading is the density of A candidates found there (lifts of the first sequence that pass the spectral filter), blue the density of B candidates. Scroll to zoom, hover for details, and click to select one or more patches to search; you can add your organization's logo, and adjacent searched squares of yours merge into one larger logo.
Explore a patch
Zoom in on the search space and select as many patches as you would like to search. A slice finds a whole pair: if yours holds one, you win that pair's share of the pool.
Launch a worker
Open dcp.live/?computeGroups=legendre153,jointhehunt on any computer or device and click Start to compute pieces of the search space. If you like the browser worker, consider running a native one: see distributive.network for DCP Worker options.
WorkExplorers
Most A candidates found
Nobody yet.
Most B candidates found
Nobody yet.
The prize pool
A share of every credit spent on this search goes into a pool for the people who find a pair. Each slice searches both sequences of its class together, so a pair is found whole, in one patch, and its award goes to that patch's explorer.
| pair found | share of the pool | status |
|---|
5% of the credits paid for completed slices; the schedule halves with each pair found. Slices from jobs that did not record a price count at the default rate. Payout mechanics are not built yet.
Pairs
No verified pair yet. Every slice joins its A and B candidates on their exact autocorrelations, and the relay checks each reported pair again; only a proven pair appears here.
How the search works
Length 153 = 9 × 17. The conjecture fixes the 9-compression of each sequence: +1 first, then −3 on the squares modulo 17 and +3 elsewhere. This search takes the family that also has the symmetry of the 117 solution: both sequences are fixed by the multiplier 118 (118 ≡ −1 mod 17, ≡ 1 mod 9). That family is small enough to search completely, in three steps:
Compress first
Fold 153 entries into 51 by adding positions 51 apart. The folded pair must already have matching spectra and autocorrelations. With the symmetry, each folded sequence is set by 9 columns: 10⁸ candidates each, which leave 28,696 compatible folded pairs in 2222 classes. This step takes seconds.
Lift, one choice at a time
Unfolding a ±1 entry of the folded sequence means choosing where its odd entry sits among 3 positions; the symmetry ties choices in pairs. Choices are walked in Gray-code order with the spectrum updated incrementally, and every lift that breaks the spectral bound is dropped on the spot.
Join exactly
Inside a slice, surviving A and B lifts are matched on their exact autocorrelations, and any match is checked as a pair. Shifts and a common multiplier that change nothing are fixed in advance, so no lift is searched twice. 11,688 slices cover about 9 × 10¹² lifts.
Each slice reports its counts and any pair to the relay, which keeps exact account of what has been searched. Searching every slice settles whether a pair with this symmetry exists. The same code, run on length 117, reproduces Blackthorn's published census and finds his pair. Code and write-up.
Why it matters
A Legendre pair of length ℓ builds a Hadamard matrix of order 2ℓ + 2: a square of +1 and −1 entries whose rows are all mutually orthogonal. Hadamard matrices are the backbone of a surprising amount of engineering. Walsh–Hadamard codes separate the channels in CDMA mobile networks; the Reed–Muller codes built from them carried Mariner 9's pictures of Mars; Hadamard masks multiplex light in spectrometers and single-pixel cameras; Plackett–Burman screening designs, which are Hadamard matrices in disguise, decide which factors matter in industrial experiments; and compressive sensing and some quantum-information constructions lean on them too. Circulant-structured Hadamard matrices, the kind a Legendre pair produces, are the practical ones: they need almost no storage and their transforms run through the FFT.
To be clear about the prize itself: Hadamard matrices of order 308 already exist (every order below 668 is known), so a pair of length 153 would give a new, structured matrix of that order, not the first one. The mathematical stake is larger than one matrix. Legendre pairs are conjectured to exist for every odd length, which would imply the Hadamard conjecture, open since 1893. For lengths nine times a prime, Kotsireas's conjecture predicts pairs with a specific compression. Its 117 case was settled in October 2026 (Blackthorn); 153 is the next open case, so finding a pair, or exhausting a clearly defined part of its space with exact bookkeeping, is new evidence either way.
Where the method goes next
The search itself is the other product: exact bookkeeping of what was covered, results that anyone can re-verify, and search code that runs in any browser. The same machinery fits sequence and design searches where better objects are used today.
Low-autocorrelation sequences
Finding the ±1 sequence of length N with the cleanest autocorrelation (the LABS problem) is a standard benchmark for quantum and quantum-inspired optimizers. Proven optima stop at N = 66, set in 2016. Each new proven length gives the field ground truth it currently lacks.
Pulse design
The same sequences shape radar and sonar pulses: low autocorrelation means a sharp echo with weak side lobes. Golay complementary pairs, a close cousin of Legendre pairs, also tame signal peaks in OFDM radios.
Covering arrays
A covering array is the smallest set of test cases that exercises every combination of a few settings. Smaller arrays mean fewer tests, and the best known tables improve through exactly this kind of exhaustive, verifiable search.
Screening designs
Plackett–Burman and related designs, built from Hadamard matrices, decide which factors matter in a manufacturing or lab experiment. Standard sizes are covered; unusual run sizes still benefit from new constructions.