Ponder This Challenge - November 2026 - The SumSum

The SumSum

Given a finite sequence of length $N$, $a_0,a_1, a_2,\ldots,a_{N-1}$, let's first remove the sequence's first element. We then define the sequence of the partial sums of its remaining elements as $a_1, a_1+a_2,a_1+a_2+a_3, \ldots ,\sum_{i=1}^{N-1} a_i$.

For this sequence of length N−1, let's repeat the "remove first element" and "generate new sequence from the partial sums" operations to obtain sequences of length $N-2, N-3, N-4, \ldots,N-(N-1)$ until we end up with a one-element sequence. We call this element The SumSum of the original sequence.

For example, given the 10-element sequence $1, 2, 3, 4, 5, 6, 7, 8, 9, 10$, the SumSum of the sequence is 16,796.

Your goal: For $N=10^{10}$ and $p=10^{10}+19$, find the SumSum modulo $p$ of the sequence $a_0,\ldots,a_{N-1}$ defined by $a_i={N \choose i}$.

A Bonus "*" will be given for finding for $N=10^{11}$ and $p=10^{11}+3$ the SumSum modulo $p$ of the sequence given by $a_i = 13^i$.

Solvers

• Daniel Chong Jyh Tar (1/11/2026 6:26 PM IDT)
• *Lazar Ilic (1/11/2026 7:58 PM IDT)
• *Paul Lupascu (1/11/2026 8:43 PM IDT)
• *Jean-François Hermant (1/11/2026 8:45 PM IDT)
• *Prashant Wankhede (1/11/2026 9:14 PM IDT)
• *Ankit Chakraborty (1/11/2026 9:24 PM IDT)
• *Nadir S. (1/11/2026 9:47 PM IDT)
• *Alper Halbutogullari (1/11/2026 11:05 PM IDT)
• *Jackson La Vallee (1/11/2026 11:23 PM IDT)
• Ahmet Yuksel & Nihal Pala (2/11/2026 12:22 AM IDT)
• *Bertram Felgenhauer (2/11/2026 1:09 AM IDT)
• Stéphane Higueret (2/11/2026 1:45 AM IDT)
• *King Pig (2/11/2026 3:55 AM IDT)
• *Guangxi Liu (2/11/2026 5:09 AM IDT)
• *Björn Richard Nerbe (2/11/2026 8:59 AM IDT)
• *Juergen Koehl (2/11/2026 10:29 AM IDT)
• *Kang Jin Cho (2/11/2026 10:59 AM IDT)
• *Lucas Reymond (2/11/2026 1:57 PM IDT)