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A229488
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Conjecturally, possible differences between prime(k)^2 and the previous prime for some k.
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2
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1, 2, 6, 8, 12, 14, 18, 20, 24, 26, 30, 32, 38, 42, 44, 48, 50, 54, 56, 60, 62, 66, 68, 72, 74, 78, 80, 84, 86, 90, 92, 96, 98, 102, 104, 108, 110, 114, 116, 120, 122, 126, 128, 132, 134, 138, 140, 146, 150, 152, 156, 158, 162, 164, 168, 170, 174, 176, 180
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OFFSET
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1,2
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COMMENTS
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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.
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.
[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.
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).
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.]
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)
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LINKS
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MATHEMATICA
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t = Table[p2 = Prime[k]^2; p2 - NextPrime[p2, -1], {k, 100000}]; Take[Union[t], 60]
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CROSSREFS
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Cf. A004277 (conjecturally, possible gaps between adjacent primes).
Cf. A054270 (prime below prime(n)^2).
Cf. A229489 (possible differences between prime(k)^2 and the next prime).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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