Emma Haruka Iwao's 2019 record held 31,415,926,535,897 digits of pi — and every one was checked without recomputing the whole thing.

When Emma Haruka Iwao computed pi to 31,415,926,535,897 digits in 2019, developer Alexander J. Yee confirmed the result by spot-checking distant digits rather than redoing all 31.4 trillion.

The number itself is a wink: 31,415,926,535,897 digits of pi, a count whose own leading figures spell out 3.1415926535897. Emma Haruka Iwao, then a Google Cloud developer advocate, reported that total in her company's blog post "Pi in the sky," published for Pi Day — March 14 — in 2019. It won her a Guinness World Record for the most accurate value of pi ever calculated, as NPR confirmed in its profile "The Woman Who Calculated 31 Trillion Digits Of Pi."

The quietly remarkable part isn't the size of the number. It's that nobody had to compute the whole thing twice to trust it.

How you verify 31 trillion digits without starting over

Iwao's computation used the Chudnovsky formula, the workhorse behind nearly every modern pi record. According to Alexander J. Yee's report "Google Cloud Topples the Pi Record" on numberworld.org, the calculation ran for 121 days across a cluster of Google Cloud virtual machines. Yee is the author of y-cruncher, the program that did the arithmetic, and the Chudnovsky method is a converging series: to produce digit 31 trillion, you have to pass through all the digits before it. There is no shortcut into the middle.

That property is exactly why a second method is needed to check the first. If you re-ran Chudnovsky and got the same answer, a hidden bug could simply reproduce itself. What you want is an entirely different route to a specific digit — one that lands on the same value only if the original run was correct.

Such routes exist. In 1995, Simon Plouffe, David Bailey, and Peter Borwein published what became known as the BBP formula, which can compute a given hexadecimal digit of pi directly, without calculating any of the digits before it. Fabrice Bellard later derived a faster seven-term variant. These "digit-extraction" formulas are the opposite of Chudnovsky: slow to produce all the digits, but able to jump to a chosen position almost instantly.

So Yee ran the check from the far end. As he wrote on numberworld.org, "Verification was done twice using Bellard's 7-term BBP and the original 4-term BBP formulas." Iwao echoed this on the Google Cloud blog: Yee "independently verified the calculation using Bellard's formula and BBP formula." Both verification runs finished in December 2018 — months before the March 2019 announcement — so the record was already confirmed when the Pi Day press arrived.

The vivid detail is the geometry of it. Chudnovsky grinds forward from the first digit; BBP reaches in from a distant hexadecimal position. The two meet at the same spot. If they agree, the long computation is trusted; if they disagreed, something upstream had gone wrong. They agreed.

Why two BBP checks isn't quite two separate algorithms

It's worth being precise about what "verified twice" means, because the publishers' own phrasing is a little loose. Bellard's seven-term formula and the original four-term BBP formula are both members of the same BBP family — both are digit-extraction formulas built on the same underlying idea. Running both does catch implementation errors, software bugs, and hardware faults, and it confirms the Chudnovsky result from a genuinely independent direction. But it is not the same as checking the answer with two mathematically unrelated algorithms. That distinction is often blurred in coverage, and it survives into the official sources here.

None of that undercuts the record. The verification was real, independent of the main computation, and completed before the announcement. It's simply a reminder that "independently verified" can mean different things, and that even a Guinness-certified world record rests on choices about what counts as an independent check.

Which leaves the open question every records watcher eventually asks: as these totals climb into the hundreds of trillions of digits, and now the hundred-trillion mark set in 2022, how many fundamentally different formulas should a record demand before we call a number proven? The record holders use what they have. Whether that's enough is a judgment, not a theorem.