Prime Numbers

308 Chapter 6 SUBEXPONENTIAL FACTORING ALGORITHMS
such special forms; but, like the mighty NFS, the notions can for the most
part be extended to more general composite n.
(1) Use the explicit congruences
258883717 2 mod M29 = −2 · 3 · 5 · 29 2 ,
301036180 2 mod M29 = −3 · 5 · 11 · 79,
126641959 2 mod M29 =2· 3 2 · 11 · 79,
to create an appropriate nontrivial congruence u 2 ≡ v 2 and thereby
discover a factor of M29.
(2) It turns out that √ 2 exists modulo each of the special numbers n = Fk,k ≥
2, and the numbers n = Mq,q ≥ 3; and remarkably, one can give explicit
such roots whether or not n is composite. To this end, show that
2 3·2k−2
− 2 2k−2
, 2 (q+1)/2
are square roots of 2 in the respective Fermat, Mersenne cases. In addition,
give an explicit, primitive fourth root of (−1) for the Fermat cases, and
an explicit ((q mod 4)-dependent) fourth root of 2 in the Mersenne cases.
Incidentally, these observations have actual application: One can now
remove any power of 2 in a squared residue, because there is now a closed
form for √ 2 k ; likewise in the Fermat cases factors of (−1) in squared
residues can be removed.
(3) Using ideas from the previous item, prove “by hand” the congruence
2(2 6 − 8) 2 ≡ (2 6 +1) 2 (mod M11),
and infer from this the factorization of M11.
(4) It is a lucky fact that for a certain ω, a primitive fourth root of 2 modulo
M43, wehave
2704ω 2 − 3 2 mod M43 =2 3 · 3 4 · 43 2 · 2699 2 .
Use this fact to discover a factor of M43.
(5) For ω a primitive fourth root of −1 modulo Fk,k ≥ 2, and with given
integers a, b, c, d, set
x = a + bω + cω 2 + dω 3 .
It is of interest that certain choices of a, b, c, d automatically give small
squares—one might call them small “symbolic squares”—for any of the
Fk indicated. Show that if we adopt a constraint
ad + bc =0
then x 2 mod Fk can be written as a polynomial in ω with degree less than
3. Thus for example
−6+12ω +4ω 2 +8ω 3 2 ≡ 4(8ω 2 − 52ω − 43),