Binomial coefficient modulo 142857

Question

How to calculate binomial coefficient modulo 142857 for large n and r. Is there anything special about the 142857? If the question is modulo

Answer 1

The algorithm is:

• factorise the base into prime powers; 142857 = 3^3×11×13×37
• compute the result modulo each prime power
• combine the results using the Chinese Remainder Theorem.

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To compute (n above k) mod p^q:

Source: http://www.dms.umontreal.ca/~andrew/PDF/BinCoeff.pdf, theorem 1

define (n!)_p as the product of numbers 1..n that are not divible by p

define n_j as n after erasing j least significant digits in base p

define r as n-k

define e_j as the number of carries when adding k+r, not counting the carries from j lowest digits, computing in base p

define s as 1 if p=2 & q>=3 and -1 otherwise

then (n above k) mod p^q := p^e_0 * s^e_(q-1) * concatenate(j=d..0)( (n_j!)_p / ((k_j!)_p*(r_j!)_p) ) with each term of the concatenation computing one base-p digit of the result, lowest j computing the least significant non-zero digits.