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Modulo 2, every integer is a quadratic residue. Modulo an odd prime number p there are (p + 1)/2 residues (including 0) and (p − 1)/2 nonresidues, by Euler's criterion.In this case, it is customary to consider 0 as a special case and work within the multiplicative group of nonzero elements of the field (/).
In his original paper, Rosa proved that an Eulerian graph with number of edges m ≡ 1 (mod 4) or m ≡ 2 (mod 4) cannot be graceful. [2] Also in his original paper, Rosa proved that the cycle C n is graceful if and only if n ≡ 0 (mod 4) or n ≡ 3 (mod 4). All path graphs and caterpillar graphs are graceful. [2]
a 1 = 20615674205555510, a 2 = 3794765361567513 (sequence A083216 in the OEIS). In this sequence, the positions at which the numbers in the sequence are divisible by a prime p form an arithmetic progression; for instance, the even numbers in the sequence are the numbers a i where i is congruent to 1 mod 3. The progressions divisible by ...
Quartic or biquadratic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x 4 ≡ p (mod q) is solvable; the word "reciprocity" comes from the form of some of these theorems, in that they relate the solvability of the congruence x 4 ≡ p (mod q) to that of x 4 ≡ q (mod p).
The main change made to produce the Hydra was the Mk. 66 motor which uses a new propellant that offers considerably more thrust, 1,335 pounds-force (5,940 N) (Mod 2/3) 1,415 pounds-force (6,290 N) (Mod 4).
We can use this fact to prove part of a famous result: for any prime p such that p ≡ 1 (mod 4), the number (−1) is a square (quadratic residue) mod p. For this, suppose p = 4k + 1 for some integer k. Then we can take m = 2k above, and we conclude that (m!) 2 is congruent to (−1) (mod p).
On June 8, 2004, United Defense (acquired by BAE Systems) received a $395.5 million contract to produce the Mod 2. [6] In March 2011, the U.S. Navy awarded a contract to BAE Systems to develop a variant of the Mod 2 with a directed energy system, referred to as the Mk 38 Mod 2 Tactical Laser System
The MOD 1 performed effectively in Navy riverine patrol craft and broader applications for the MK 19 were found. In 1973 the Navy developed the MOD 2, which featured improved reliability, safety, and maintainability. In 1976 a complete redesign resulted in the MK 19, MOD 3, which the Army adopted in 1983.
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