A099332 -id:A099332 - cp's OEIS frontend

A100270 Smallest odd prime of the form x^2^n + y^2^n such that x^2^k + y^2^k is prime for k=0,1,...,n-1.

3, 5, 17, 257, 65537, 43969786939269621239851427694879659964972193373572605276547046131629468448105886917662485986957414531083768961

Offset: 0

T. D. Noe, Nov 11 2004

The first five terms are the Fermat primes A019434, which are obtained with x=1 and y=2. Can a solution {x,y} be found for any n? The Mathematica program, for each n, generates numbers of the form x^2^n + y^2^n in order of increasing magnitude; it stops when all the x^2^k + y^2^k are prime for k=0,...,n.

			a(5) = 720^32+2669^32 is prime, as are 720^16+2669^16, 720^8+2669^8, 720^4+2669^4, 720^2+2669^2 and 720+2669.

Eric Weisstein's World of Mathematics, Generalized Fermat Number.

Cf. A099332, A100268, A100269.

Table[pwr=2^n; xmax=2; r=Range[xmax]+1; num=(r-1)^pwr+r^pwr; While[p=Min[num]; x=Position[num, p][[1, 1]]; y=r[[x]]; r[[x]]++; num[[x]]=x^pwr+r[[x]]^pwr; If[x==xmax, xmax++; AppendTo[r, xmax+1]; AppendTo[num, xmax^pwr+(xmax+1)^pwr]]; allPrime=True; k=0; While[k<=n&&allPrime, allPrime=PrimeQ[x^2^k+y^2^k]; k++ ]; !allPrime]; p, {n, 0, 5}]

A100268 Primes of the form x^4 + y^4 with x^2 + y^2 and x+y also prime.

Original entry on oeis.org

2, 17, 97, 257, 641, 1297, 4177, 4721, 12401, 15937, 16561, 38561, 65537, 83537, 89041, 105601, 140321, 160081, 204481, 283937, 284881, 384817, 391921, 411361, 462097, 471617, 531457, 643217, 824641, 838561, 1049201, 1089841, 1342897

Offset: 1

T. D. Noe, Nov 11 2004

nonn

The first Mathematica program generates numbers of the form x^4 + y^4 in order of increasing magnitude; it accepts a number when all the x^2^k + y^2^k are prime for k=0,1,2.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Generalized Fermat Number

Cf. A099332, A100269, A100270.

n=2; pwr=2^n; xmax=2; r=Range[xmax]; num=r^pwr+r^pwr; Table[While[p=Min[num]; x=Position[num, p][[1, 1]]; y=r[[x]]; r[[x]]++; num[[x]]=x^pwr+r[[x]]^pwr; If[x==xmax, xmax++; AppendTo[r, xmax+1]; AppendTo[num, xmax^pwr+(xmax+1)^pwr]]; allPrime=True; k=0; While[k<=n&&allPrime, allPrime=PrimeQ[x^2^k+y^2^k]; k++ ]; !allPrime]; p, {40}]
With[{nn=40},Select[Union[Transpose[Select[Total/@{#^4,#^2,#}&/@ Tuples[ Range[nn],2],AllTrue[#,PrimeQ]&]][[1]]],#<=nn^4+1&]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 23 2015 *)

A100269 Primes of the form x^8 + y^8 with x^4 + y^4, x^2 + y^2 and x+y also prime.

Original entry on oeis.org

2, 257, 65537, 2724909545357921, 3282116715437377, 40213879071634241, 147578912575757441, 303879829574456257, 697576026529536481, 1316565220482548321, 2860283484326400961, 4080251077774711937

Offset: 1

T. D. Noe, Nov 11 2004

nonn

The Mathematica program generates numbers of the form x^8 + y^8 in order of increasing magnitude; it accepts a number when all the x^2^k + y^2^k are prime for k=0,1,2,3.

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Eric Weisstein's World of Mathematics, Generalized Fermat Number.

Cf. A099332, A100268, A100270.

n=3; pwr=2^n; xmax=2; r=Range[xmax]; num=r^pwr+r^pwr; Table[While[p=Min[num]; x=Position[num, p][[1, 1]]; y=r[[x]]; r[[x]]++; num[[x]]=x^pwr+r[[x]]^pwr; If[x==xmax, xmax++; AppendTo[r, xmax+1]; AppendTo[num, xmax^pwr+(xmax+1)^pwr]]; allPrime=True; k=0; While[k<=n&&allPrime, allPrime=PrimeQ[x^2^k+y^2^k]; k++ ]; !allPrime]; p, {20}]

A182288 Primes of form a^2+b^2 such that |a|+|b| is prime of the form 4k+3.

Original entry on oeis.org

5, 29, 37, 61, 73, 101, 181, 193, 241, 269, 277, 293, 349, 409, 521, 541, 593, 661, 701, 929, 937, 1009, 1069, 1109, 1117, 1129, 1217, 1237, 1249, 1289, 1609, 1741, 1753, 1789, 1801, 2029, 2053, 2161, 2221, 2269, 2357, 2389, 2521, 2557, 2609, 2633, 2741, 2753

Offset: 1

Thomas Ordowski, Apr 23 2012

nonn

			The prime 29=5^2+2^2 such that 5+2=7 is prime of form 4k+3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A099332.

nn = 60; t = {}; Do[p1 = a^2 + b^2; p2 = a + b; If[p1 < nn^2 && PrimeQ[p1] && Mod[p2, 4] == 3 && PrimeQ[p2], AppendTo[t, p1]], {a, nn}, {b, a}]; Sort[t] (* T. D. Noe, Apr 23 2012 *)

list(lim)=my(v=List(),t);for(a=1,sqrt(lim),forstep(b=(2-a)%4+1, min(a,sqrt(lim-a^2)),4,if(isprime(a+b)&&isprime(t=a^2+b^2), listput(v,t)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Apr 24 2012

Extended by T. D. Noe, Apr 23 2012

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A100270 Smallest odd prime of the form x^2^n + y^2^n such that x^2^k + y^2^k is prime for k=0,1,...,n-1.

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A100268 Primes of the form x^4 + y^4 with x^2 + y^2 and x+y also prime.

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A100269 Primes of the form x^8 + y^8 with x^4 + y^4, x^2 + y^2 and x+y also prime.

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A182288 Primes of form a^2+b^2 such that |a|+|b| is prime of the form 4k+3.

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