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A129830
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Conjectured first occurrence of numbers n with the property that there exist two consecutive primes p and q such that pq + n is a fourth power.
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0
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1, 4, 9, 10, 25, 35, 36, 46, 49, 64, 66, 75, 113, 144, 149, 179, 188, 196, 203, 221, 241, 250, 290, 302, 380, 395, 397, 400
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OFFSET
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1,2
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COMMENTS
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It remains to prove that for certain n, pq+n != y^4 for all consecutive primes p and q. This list was computed for p and q with prime indices up to 10000.
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LINKS
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EXAMPLE
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p=7,q=11,k=4. 7*11+4 = 81 = 3^4.
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PROG
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(PARI) primefourth(n, m) = { local(c, k, x, p1, p2, j); c=0; for(k=1, m, for(x=1, n, p1=prime(x); p2=(prime(x+1)); y=p1*p2+k; if(isfourth(y), c++; print1(k", "); break; ) ) ); c; } isfourth(n) = \Return 1 if n is a fourth power { local(r); r = sqrt(sqrt(n)); if(floor(r+.5)^4== n, 1, 0) }
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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