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TAP TIMES -27- DECEMBER 2022
CalTech solves a 175 When Gauss was young, he amazed his
year-old math problem classmates by quickly developing a formu-
la for adding up the numbers 1 to 100.
Pasadena, Calif. —For the past 175
years, a perplexing feature of num- Distribution of solutions
bers first discovered by Ernst Kum- Gauss later developed a complex concept
mer, a German mathematician, has known as Gauss sums, which readily map
confounded researchers. the distribution of solutions to equations.
Number theory He looked at the distribution of what are
called square Gauss sums for nontrivial
At one point in the 1950s, this quirky prime numbers (primes that have a re-
feature of number theory was thought to mainder of 1 when you divide by 3) and
have been wrong, but then, decades later, found a “beautiful structure,” according to
mathematicians found hints that it was in Radziwill.
fact true.
This summing activity involves a type of
Now, after several twists and turns, two math known as modular arithmetic. An
Caltech mathematicians have at last found easy way to understand modular arithme-
proof that Kummer was right all along. tic is to think of a clock and its face divid-
ed into 12 hours.
“We had several ‘aha’ moments, but then
you have to roll up your sleeves and figure When noon or midnight rolls around, the
this out,” explains Alexander (Alex) Dunn, numbers are reset and go back to 1. This
a postdoc at Caltech who wrote the proof “modulo 12” system simplifies timekeep-
with his advisor, professor of mathematics ing, since we do not have to keep counting
Maksym Radziwill and posted it online in up hours forever.
September 2021.
Making equations simpler
Gauss sums In the case of Gauss sums, the same idea
The math problem has to do with Gauss is at play but the base “clock face” is di-
sums, which are named after the 18th- vided up into p hours, where p is a prime
century prolific mathematician Carl Fried- number. “Modulo p math is a way of strip-
rich Gauss. ping out information and making impossi-
bly complicated equations simpler,” Radzi-
will says.
Mathematician Samuel Patterson who
proposed a solution to the mix-up in 1978,
now referred to as Patterson’s conjecture.
He was a graduate student at the Univer-
sity of Cambridge at the time and recog-
nized that the bias in the distribution of
the solutions could be overwhelmed as the
sample size gets bigger and bigger.
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CalTech researchers (left) Maksym Radziwill and Alexander Dunn
