with a , b any coprime integers, a > 1 and − a < b < a . (Since a n − b n is always divisible by a − b , the division is necessary for there to be any chance of finding prime numbers. In fact, this number is the same as the Lucas number U n ( a + b , ab ) , since a and b are the roots of the quadratic equation x 2 − ( a + b ) x + ab = 0 , and this number equals 1 when n = 1 ) We can ask which n makes this number prime. It can be shown that such n must be primes themselves or equal to 4, and n can be 4 if and only if a + b = 1 and a 2 + b 2 is prime. (Since a 4 − b 4 / a − b = ( a + b )( a 2 + b 2 ) . Thus, in this case the pair ( a , b ) must be ( x + 1, − x ) and x 2 + ( x + 1) 2 must be prime. That is, x must be in A027861 .) It is a conjecture that for any pair ( a , b ) such that for every natural number r > 1 , a and b are not both perfect r th powers, and −4 ab is not a perfect fourth power . there are infinitely many values of n such that a n − b n / a − b is prime. (When a and b are both perfect r th powers for an r > 1 or when −4 ab is a perfect fourth power, it can be shown that there are at most two n values with this property, since if so, then a n − b n / a − b can be factored algebraically) However, this has not been proved for any single value of ( a , b ) .
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The DOTM packaging release of this 2-pack features an altered paint application on Optimus Prime's ion blaster/barrage cannon. On the original ROTF packaging release, the slide-out barrel on Prime's blaster is painted silver with a black paint spray app to simulate usage of the weapon, and the exposed inside wall of the gas tank at the top of the blaster is left as unpainted grey plastic (exactly like the blaster on the original single-boxed ROTF Buster Optimus Prime). After receiving numerous reports from customers that the silver paint on the slide-out barrel was easily scraped off by merely transforming it from gas tank to blaster, Takara-Tomy briefly stated in an interview that they had addressed the issue for the Dark of the Moon packaged version by swapping the placement of the silver paint application. Thus, the Dark of the Moon packaged version now features the slide-out barrel on Prime's blaster as unpainted grey plastic with an unchanged black paint spray app to simulate usage of the weapon, and the exposed inside wall of the gas tank at the top of the blaster is now painted silver.