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A322269 a(n) is the largest minimal prime P such that, for any odd number b, the product P*b is a nonzero square modulo 8 and modulo each of the first n primes. 7
7, 23, 83, 311, 1873, 3583, 12289, 33049, 67369, 174241, 552841, 1010881, 3267289, 7921489, 12537719, 30706079, 56988649, 108345169, 328583161, 880051561, 1644946249 (list; graph; refs; listen; history; text; internal format)



When factoring a number b using the quadratic sieve, it can be practical to multiply b by a certain factor f so that the product f*b is a square modulo several small primes. It is desirable that f be prime, because the prime factors of f cannot be used in the factor base of the quadratic sieve.

To find such an f for a given b and the first n primes, it must be checked whether b is a square or not, modulo each of these primes. Then f is the smallest prime (or 1) which satisfies the same conditions, modulo each of these primes.

Letting p=prime(n), an f can be found for each of the possible values of b (mod p#, the primorial of p), coprime to p#. (Actually we are using a period of 4*(p#), because instead of mod 2 we check for mod 8.) a(n) is the largest of all these values of f.

8 was chosen instead of 2, because there is a unique quadratic residue (mod 8), i.e., 1, for all odd numbers.

Sequences A322271 to A322275 are separate listings for the sequences of all f, corresponding to n=2 to 6, which illustrate the idea further.

For finding the full sequences of all f, instead of checking all b mod 4*(p#), it is more practical to check all prime numbers (and including 1) in order, whether they are suitable as an f or not. Each prime receives a "code" of Boolean flags which indicate whether it is a square or not, modulo each of the first n primes. If it is the first prime with this specific "code", then every value of b mod 4*(p#) which has the same "code" is assigned this prime as its f. This process is repeated until all possible "codes" have an f assigned. (The flag for mod 8, instead of only signaling "is (not) a square", has four different values: 1, 3, 5, and 7.)

A322270(n) is the code corresponding to a(n).

In order to satisfy the conditions, both f and b must be coprime to p#, i.e., f must either be 1 or greater than prime(n).


Table of n, a(n) for n=1..21.


For n=3, we want the product to be a square mod 8, mod 2, mod 3 and mod 5. The corresponding products b*f are, for all b < 120 and coprime to 120:

1*1, 7*7, 11*11, 13*13, 17*17, 19*19, 23*23, 29*29, 31*31, 37*13, 41*41, 43*43, 47*23, 49*1, 53*53, 59*11, 61*61, 67*43, 71*71, 73*73, 77*53, 79*31, 83*83, 89*41, 91*19, 97*73, 101*29, 103*7, 107*83, 109*61, 113*17, 119*71. (See A322272.)

The largest f in this set is 83 (associated with b=83 and b=107). Therefore a(3) = 83.



QresCode(n, nPrimes) = {

  code = bitand(n, 7)>>1;

  for (j=2, nPrimes,

    x = Mod(n, prime(j));

    if (issquare(x), code += (1<<j));


  return (code);


a322269(n) = {

  totalEntries = 1<<(n+1);

  f = vector(totalEntries);

  f[totalEntries-3] = 1;  \\ 1 has always the same code: ...111100

  counter = 1;

  forprime(p=prime(n+1), +oo,

    code = QresCode(p, n);

    if (f[code+1]==0,


      counter += 1;

      if (counter==totalEntries, return(p));





Binary codes as described above are given in A322270.

Sequences for all f associated with a specific n are given in A322271 (n=2), A322272 (n=3), A322273 (n=4), A322274 (n=5), and A322275 (n=6).

Sequence in context: A086908 A093069 A341665 * A303890 A003540 A063793

Adjacent sequences:  A322266 A322267 A322268 * A322270 A322271 A322272




Hans Ruegg, Dec 01 2018



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Last modified June 15 12:29 EDT 2021. Contains 345048 sequences. (Running on oeis4.)