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A229488 Conjecturally, possible differences between prime(k)^2 and the previous prime for some k. 2


%S 1,2,6,8,12,14,18,20,24,26,30,32,38,42,44,48,50,54,56,60,62,66,68,72,

%T 74,78,80,84,86,90,92,96,98,102,104,108,110,114,116,120,122,126,128,

%U 132,134,138,140,146,150,152,156,158,162,164,168,170,174,176,180

%N Conjecturally, possible differences between prime(k)^2 and the previous prime for some k.

%C Are there any missing terms? The first 10^7 primes were examined. All these differences occur for some k < 10^5. Note that the first differences of these terms is 1, 2, 4, or 6.

%C From _R. J. Mathar_, Oct 29 2013: (Start)

%C This sequence of possible differences d= prime(k)^2 -q looks similar to A047238; 1 is an exception associated with the single even prime, 1=2^2-3.

%C [Reason: Otherwise primes are odd, squared primes are also odd, so the differences are even and therefore in the class {0,2,4} mod 6.

%C Furthermore primes are of the form 3n+1 or 3n+2, squared primes are of the form 9n^2+6n+1 or 9n^2+12n+4, so squared primes are of the form ==1 (mod 3).

%C The difference prime(k)^2-q is therefore the difference between a number ==1 (mod 3) and a number == {1,2} (mod 3) and therefore a number == {0,2} mod 3. This is never of the form 6n+4 ( == 1 mod 3). So the differences are in the class {0,2} mod 6, demonstrating that this is essentially a subsequence of A047238.]

%C Furthermore, differences 36, 144, 324,... of the form (6n)^2, A016910, appear in A047238 but not here, because prime(k)^2 -q=(6n)^2 is equivalent to prime(k)^2-(6n)^2 =q =(prime(k)+6n)*(prime(k)-6n), which requires an explicit factorization of the prime q. This is a contradiction if we assure that prime(k)-6n is not equal 1; if we scanned explicitly all primes up to prime(k)=10^7, for example, all (6n)^2 up to 6n<=10^7 are proved not to be in the sequence. (End)

%t t = Table[p2 = Prime[k]^2; p2 - NextPrime[p2, -1], {k, 100000}]; Take[Union[t], 60]

%Y Cf. A000040 (primes), A001248 (primes squared).

%Y Cf. A004277 (conjecturally, possible gaps between adjacent primes).

%Y Cf. A054270 (prime below prime(n)^2).

%Y Cf. A229489 (possible differences between prime(k)^2 and the next prime).

%K nonn

%O 1,2

%A _T. D. Noe_, Oct 21 2013

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