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Squares equal to the difference between two successive primes of the form n^2+1.
+10
5
64, 144, 100, 1024, 4900, 10816, 11664, 12544, 18496, 102400, 41616, 46656, 331776, 298116, 44100, 451584, 270400, 141376, 372100, 678976, 504100, 1849600, 524176, 2890000, 3504384, 602176, 685584, 8702500, 1768900, 2160900, 868624, 532900, 624100, 12960000
EXAMPLE
64 is in the sequence because 6^2 + 1 = 37, 10^2+1 = 101 and 101 - 37 = 64 is square.
MAPLE
q:=2:for n from 2 to 100 do:p:=n^2+1:if type(p, prime)=true then x:=p-q:q:=p: z:=sqrt(x):if z=floor(z) then printf(`%d, `, z):else fi:od:
MATHEMATICA
Select[#[[2]]-#[[1]]&/@Partition[Select[Range[2000000]^2+1, PrimeQ], 2, 1], IntegerQ[ Sqrt[#]]&] (* Harvey P. Dale, Nov 08 2017 *)
Least k such that p = k^2 + 1 and q = (k+2n)^2 + 1 are prime numbers with q - p square.
+10
2
24, 6, 312984, 16896, 120, 734994, 10640, 10, 1946016, 150, 171864, 180, 31200, 17136, 120, 84, 8976, 54, 137256, 300, 231504, 66, 184, 360126, 24, 5824, 2496, 224, 261696, 90, 4359344, 66, 50160, 68816, 280, 864, 1524696, 570, 219336, 11520, 8487984, 126, 22704
COMMENTS
4*n*(k + n) is a square. If n is a square, then k + n is also a square.
If n is prime, then n divides k.
If we add the additional condition that p and q are two consecutive primes of the form m^2 + 1, then we obtain the sequence A339008, with A339008(n) = a(n) for n = 1, 2, 3, 4, 6, 7 and 9.
EXAMPLE
a(1) = 24 because 24^2 + 1 = 577, (24 + 2)^2 + 1 = 677 and 677 - 577 = 10^2 is a square. The other values m such that p = m^2 + 1 and q = (m+2)^2 + 1 are primes with q - p square are 11024, 133224, 156024, 342224, 416024,...
a(2) = 6 because 6^2 + 1 = 37, (6 + 4)^2 + 1 = 101 and 101 - 37 = 8^2 is a square. The other values m such that p = m^2 + 1 and q = (m+4)^2 + 1 are primes with q - p square are 16, 126, 1350, 1456, 1566, 2310, 5200,...
MAPLE
for n from 1 to 50 do:
ii:=0:
for k from 2 by 2 to 10^9 while(ii=0) do:
p:=k^2+1:q:=(k+2*n)^2 +1:
if isprime(p) and isprime(q) and sqrt(q-p)=floor(sqrt(q-p))
then
ii:=1:printf(`%d %d \n`, n, k):
else
fi:
od:
od:
PROG
(PARI) a(n) = my(k=1); while (!(isprime(p=k^2+1) && isprime(q=(k+2*n)^2 + 1) && issquare(q-p)), k++); k; \\ Michel Marcus, Nov 18 2020
Least k such that p = k^2 + 1 and q = (k+2n)^2 + 1 are two consecutive prime numbers of the same form with q - p square.
+10
2
24, 6, 312984, 16896, 240, 734994, 10640, 10360, 1946016, 2550, 13189264, 72996, 416520, 2184336, 1584360, 202484, 232696, 1700150, 2394456, 375360, 8736504, 9237866, 53629744, 360126, 87000, 574339974, 82404216, 23237760, 1249877496, 826650, 127119344, 1527720
COMMENTS
4*n*(k + n) is a square. If n is a square, then k + n is also a square.
If n is prime, then n divides k.
a(n) = A339007(n) for n = 1, 2, 3, 4, 6, 7 and 9.
EXAMPLE
a(1) = 24 because 24^2 + 1 = 577, (24 + 2)^2 + 1 = 677. The numbers 577 and 677 are two consecutive primes of the form m^2+1, and 677 - 577 = 10^2 is a square. The other values m such that p = m^2 + 1 and q = (m+2)^2 + 1 are consecutive primes with q - p square are 11024, 133224, 156024, 342224, 416024, ...
a(2) = 6 because 6^2 + 1 = 37, (6 + 4)^2 + 1 = 101. The numbers 37 and 101 are two consecutive primes of the form m^2+1, and 101 - 37 = 8^2 is a square. The other values m such that p = m^2 + 1 and q = (m+4)^2 + 1 are consecutive primes with q - p square are 16, 126, 1350, 1456, 1566, 2310, 5200, ...
MAPLE
for n from 1 to 25 do:
ii:=0:n1:=0:q:=2:
for k from 2 by 2 to 10^9 while(ii=0) do:
p:=k^2+1:
if isprime(p)
then
x:=p-q:q:=p:z:=sqrt(x):
if z=floor(z) and k-n1=2*n
then
ii:=1:printf(`%d %d \n`, n, n1):
else
n1:=k:
fi:
fi:
od:
od:
PROG
(PARI) consecutive(p, q) = {forprime(r = nextprime(p+1), precprime(q-1), if (isprime(r) && issquare(r-1), return(0)); ); return(1); }
a(n) = my(k=1); while (!(isprime(p=k^2+1) && isprime(q=(k+2*n)^2 + 1) && issquare(q-p) && consecutive(p, q)), k++); k; \\ Michel Marcus, Nov 30 2020
2, 8, 10, 25, 42, 147, 160, 169, 238, 260, 491, 544, 869, 890, 923, 1140, 1337, 1386, 1465, 1643, 1927, 3371, 4614, 5038, 5086, 5225, 5832, 5909, 5995, 7118, 7157, 8540, 9859, 12543, 13505, 13795, 13841, 14211, 15347, 17079, 17263, 18643, 20211, 21184, 21245
COMMENTS
The prime numbers of the sequence are 2, 491, 3371, 9859, 13841,...
The corresponding squares A002496(n) mod A002496 (n-1) are : {1, 144, 100, 1024, 4900, 10816, 11664, 12544,...} = {1} union { A216330} minus {64}.
MAPLE
with(numtheory):nn:=360000:T:=array(1..nn):kk:=0:
for n from 1 to nn do:
if type(n^2+1, prime)=true then
kk:=kk+1:T[kk]:=n^2+1:
else
fi:
od:
for m from 1 to kk-1 do:
r:=irem(T[m+1], T[m]):z:=sqrt(r):
if z=floor(z)
then printf(`%d, `, m+1):
else
fi:
od:
MATHEMATICA
lst={}; lst1={}; nn=400000; Do[If[PrimeQ[n^2+1], AppendTo[lst, n^2+1]], {n, 1, nn}]; nn1:=Length[lst];
Do[If[IntegerQ[Sqrt[Mod[lst[[m]], lst[[m-1]]]]], AppendTo[lst1, m]], {m, 2, nn1}]; lst1
PROG
(Python)
from gmpy2 import t_mod, is_square, is_prime
for n in range(1, 10**7):
....m += 2*n+1
....if is_prime(m):
........if is_square(t_mod(m, A002496_list[-1])):
Pairs of primes (p,q) = ( A002496(m), A002496(m+1)) such that q-p is a power r of the product of its prime factors for some m.
+10
0
37, 101, 577, 677, 15877, 16901, 57601, 62501, 33988901, 34035557, 113209601, 113507717, 121528577, 121572677, 345960001, 346332101, 635040001, 635544101, 7821633601, 7823402501, 17748634177, 17749167077, 24343488577, 24344112677, 97958984257, 97962740101
COMMENTS
The corresponding sequence of numbers q - p is a subsequence of A076292. Conjecture: the sequence is infinite.
The corresponding powers r are given by the sequence b(n) = 6, 2, 10, 2, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ... It seems that b(n) = 2 for n > 5.
EXAMPLE
The pair (257, 401) = (16^2+1, 20^2+1) is not in the sequence because 401 - 257 = 144 = 2^4*3^2.
The pair (577, 677) = (24^2+1, 26^2+1) is in the sequence because 577 - 677 = 100 = 2^2*5^2.
The pair (33988901, 34035557) = (5830^2+1, 5834^2+1) is in the sequence because 33988901 - 34035557 = 46656 = 2^6*3^6.
MAPLE
with(numtheory):
T:=array(1..26):nn:=350000:q:=5:j:=1:
for n from 4 by 2 to nn do:
p:=n^2+1:
if type(p, prime)=true
then
x:=p-q:r:=q:q:=p:
u:=factorset(x):n0:=nops(u):ii:=0:d:=product(u[i], i=1..n0):
for k from 2 to 20 while(ii=0) do:
if d^k=x
then ii=1:T[j]:=r:T[j+1]:=q:j:=j+2:
else
fi:
od:
fi:
od:
print(T):
PROG
(PARI) lista(nn) = my(lastp=2); forprime(p=nextprime(lastp+1), nn, if (issquare(p-1), if (ispowerful(p-lastp), my(f=factor(p-lastp)[, 2]); if (vecmin(f) == vecmax(f), print1(lastp, ", ", p, ", ")); ); lastp = p; ); ); \\ Michel Marcus, Feb 03 2022
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