If m is of the form 2 j , this implies a maximum period for any a of m/4 , achieved when MemberQ[{3, 5}, Mod[a, 8]] . … When m is a prime, this implies that the period can then be as long as (m - 3)/2 .

Measurements done on a binary pulsar system are nevertheless consistent at a 10 -3 level with the emission of gravitational radiation in a fairly strong gravitational field at the rate implied by general relativity.

In both cases the limited number of digits implies behavior that ultimately repeats—but only long after the other effects we discuss have occurred.)

In the early 1600s the concept of inertia developed by Galileo implied that space must have a certain fundamental uniformity.

And when effectively restricted to a finite region, this equation allowed only certain modes, corresponding to discrete quantum states—whose properties turned out to be exactly the same as implied by matrix mechanics. … But despite a few results about large-distance analogs of renormalizability, the question of what QCD might imply for processes at larger distances could not really be addressed by such methods.

Note that the existence of universal Diophantine equations implies that any problem of mathematics—even, say, the Riemann Hypothesis—can in principle be formulated as a question about the existence of solutions to a Diophantine equation.

But the standard formalism of quantum theory (see page 1058 ) implies that for an analogous quantum system—like a line of n quantum spins each either up or down—one instead has to give a whole vector of probability amplitudes for each of the 2 n possible complete underlying spin configurations.

(For quadratic equations Hasse's Principle implies that if no solutions exist for any n then there are no solutions for ordinary integers—but a cubic like 3x 3 + 4 y 3 + 5 z 3  0 is a counterexample.)

But for smaller e[s] one can show that Abs[m[s]]  (1 - Sinh[2 β ] -4 ) 1/8 where β can be deduced from e[s]  -(Coth[2 β ](1 + 2 EllipticK[4 Sech[2 β ] 2 Tanh[2 β ] 2 ] (-1 + 2 Tanh[2 β ] 2 )/ π )) This implies that just below the critical point e 0 = - √ 2 (which corresponds to β = Log[1 + √ 2 ]/2 ) Abs[m] ~ (e 0 - e) 1/8 , where here 1/8 is a so-called critical exponent.

But to reproduce the kind of exact reduction of probability amplitudes that is implied by the standard formalism of quantum theory inevitably requires taking the limit of an infinite system.