Research in mathematics and finance
We find new results on unsolved problems in mathematics, and we model the interest rates that mortgages and company bonds are priced from. Two pieces of that work play below as films, and each comes with a button that puts your own device to work on it.
Five new terms for Erdős problem 382
The new terms
| Length | Where the first stretch starts | Its largest prime |
|---|---|---|
| 10 | 4,928,180,396 | 70,201 |
| 11 | 40,533,366,231 | 201,329 |
| 12 | 111,460,496,439 | 333,857 |
| 13 | 436,502,026,483 | 660,683 |
| 14 | 17,952,448,643,078 | 4,237,033 |
The OEIS entry counts each stretch by its width, one less than its length, so these are its terms a(9) to a(13). Searching on to 200 trillion found the first stretch of sixteen, starting at 52,629,641,927,627, and no stretch of exactly fifteen. The entry records both.
Paul Erdős left hundreds of problems unsolved when he died in 1996. Number 382 is about stretches of consecutive whole numbers whose product is divisible by the square of its own largest prime factor. One of its questions is whether such a stretch can be as long as you like. The problem's sequence in the OEIS, the reference mathematicians use for integer sequences, records where the first stretch of each length begins, counting only stretches that cannot be made any longer. It had the answer for every length up to nine, and only upper bounds for lengths ten to fourteen.
We showed that all five bounds are the exact answers. Two programs, written in different ways, searched every number up to 20 trillion and agreed on every stretch. We then carried the search on to 200 trillion and found the first stretch of sixteen. The OEIS published the five new terms on 12 October 2026.
A third program, which shares no arithmetic with the other two, then checked every number up to 20 trillion again and found every stretch of ten or more that they had found, and no others.
Find the first nine on this device
The nine terms published before ours all start below 100 million. This page finds them with the third program's method, on your own computer or phone, in a few seconds.
| Length | Published | Found here |
|---|---|---|
| 1 | 4 | not yet |
| 2 | 8 | not yet |
| 3 | 48 | not yet |
| 4 | 1,680 | not yet |
| 5 | 76,725 | not yet |
| 6 | 332,925 | not yet |
| 7 | 7,474,752 | not yet |
| 8 | 4,541,154 | not yet |
| 9 | 75,047,565 | not yet |
The US Treasury yield curve, fitted for every trading day since 1990
Every inversion since 1990
| From | To | Trading days |
|---|---|---|
| 7 July 2000 | 19 January 2001 | 135 |
| 17 July 2006 | 29 May 2007 | 219 |
| 13 May 2019 | 10 October 2019 | 106 |
| 31 January 2020 | 2 March 2020 | 21 |
| 18 October 2022 | 12 December 2024 | 539 |
| 26 February 2025 | 7 May 2025 | 50 |
| 4 June 2025 | 15 September 2025 | 71 |
Each spell runs from the first to the last trading day of an inversion that lasted at least 20 trading days, with breaks of under ten days counted in. From our run over the published rates.
The yield curve shows what the United States government pays to borrow for anything from a month to thirty years. Mortgage rates and company bonds are priced from it, and the Treasury publishes its rates for every trading day.
For each of the 9,200 trading days since 2 January 1990, we fitted one smooth curve to that day's published rates. We used the Svensson model, which the European Central Bank uses for the euro area curves it publishes, and which Federal Reserve Board staff use for their own curve of US Treasury yields. The typical gap between our curve and the published rates is 2.9 basis points, where a basis point is one hundredth of one per cent.
We can use a curve like this to model a business's own loans, for instance to compare a fixed rate with a variable one.
Most of the time, borrowing for longer costs more, and the curve slopes upward. When short-term rates rise above long-term ones, the curve inverts. Beneath the curve, the film tracks the ten-year rate minus the three-month rate, in amber whenever it falls below zero, with the four recessions since 1990 shaded. The longest unbroken inversion since 1990 lasted 534 trading days, from 25 October 2022 to 12 December 2024.
Fit every day on this device
The button below downloads our copy of the Treasury's published rates for all 9,200 days, about 154 KB to download, and fits every day's curve again in your browser.
Three formal statements in Google DeepMind's library
Lean is a programming language in which a computer checks every step of a mathematical argument. Google DeepMind keeps an open library of unsolved problems stated in Lean, as a test for programs that try to prove theorems. Its maintainers review every addition before it is merged.
We wrote the statement of Erdős problem 430. Two statements already in the library, for problems 11 and 873, asked about every case where the original problems ask only about large numbers, and we corrected both. All three changes were merged on 8 October 2026.
Published results
- 12 October 2026Five new terms for Erdős problem 382OEIS A388850
- 8 October 2026The statement of Erdős problem 430, and corrections to problems 11 and 873Google DeepMind, formal-conjectures
- 7 October 2026Two new terms for Erdős problem 389OEIS A375071
- 7 October 20262,300 new terms for Erdős problem 1109OEIS A392164
- 5 October 2026Twenty-six new terms for Erdős problem 1109OEIS A392165
Tell us what you want built
A few sentences about the system you need are enough to start. We will reply with our questions, then a plan and a price.
Or write to hello@stratumics.com