998,244,353
998,244,353 is a prime equal to 119 × 2²³ + 1, used where fast polynomial multiplication is needed.
- Odd
- Prime
Where 998,244,353 fits
Why 998,244,353 is special
The structure is the point. A prime one more than a large power of two has a root of unity of that order, which is exactly what a number-theoretic transform needs — the integer equivalent of a fast Fourier transform, with no floating-point error at all. That is why it is chosen over the rounder 1,000,000,007 whenever convolution is involved.
998,244,353 is part of
Factors & digit properties
- Divisors (2)
- 1, 998244353
- Prime factorisation
- 998244353
- Digit sum
- 47
- Digital root
- 2
Representations
- Decimal
- 998244353
- Scientific notation
- 9.982 × 10^8
Roman numerals
Roman numerals as modelled here goes up to 3,999.
Binary
1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
1 × 536,870,912 + 1 × 268,435,456 + 1 × 134,217,728 + 0 × 67,108,864 + 1 × 33,554,432 + 1 × 16,777,216 + 1 × 8,388,608 + 0 × 4,194,304 + 0 × 2,097,152 + 0 × 1,048,576 + 0 × 524,288 + 0 × 262,144 + 0 × 131,072 + 0 × 65,536 + 0 × 32,768 + 0 × 16,384 + 0 × 8,192 + 0 × 4,096 + 0 × 2,048 + 0 × 1,024 + 0 × 512 + 0 × 256 + 0 × 128 + 0 × 64 + 0 × 32 + 0 × 16 + 0 × 8 + 0 × 4 + 0 × 2 + 1 × 1 = 998,244,353
Octal
7 3 4 0 0 0 0 0 0 1
7 × 134,217,728 + 3 × 16,777,216 + 4 × 2,097,152 + 0 × 262,144 + 0 × 32,768 + 0 × 4,096 + 0 × 512 + 0 × 64 + 0 × 8 + 1 × 1 = 998,244,353
Hexadecimal
3 b 8 0 0 0 0 1
3 × 268,435,456 + b × 16,777,216 + 8 × 1,048,576 + 0 × 65,536 + 0 × 4,096 + 0 × 256 + 0 × 16 + 1 × 1 = 998,244,353
Other mathematical patterns
- Deficient number
Technology & computing
The prime used where fast polynomial multiplication is needed is 998,244,353. Equal to 119 × 2²³ + 1. The large power of two in that factorisation is the point: it guarantees roots of unity of every order up to 2²³, which is what a number-theoretic transform requires and 1,000,000,007 cannot supply.
DocumentedTechnology & computingGeography: worldwide
Common questions
Is 998,244,353 a prime number?
Yes, 998,244,353 is prime — it has exactly two positive divisors: 1 and itself.
What are the factors of 998,244,353?
The factors of 998,244,353 are 1, 998244353.
About this page's data
Mathematical facts:computed directly from the number's value — nothing to source or verify externally.
Contextual records: 1 reviewer-confirmed record, sourced and reviewed before publication.
Last reviewed: 15 July 2026