OFFSET
1,1
COMMENTS
From Chai Wah Wu, May 17 2021: (Start)
Sequence is infinite.
If a, b > 1 and b^a-b == 0 mod a+1 then b^c+b+c is a term for c = ab(b^(a-1)-1)/(a+1), y = c/a, x = b^a.
If b > 1 and b <> 2 mod 3, then b^(2b(b-1)/3)+b(2b+1)/3 is a term.
If b > 2, then b^((b-1)(b^(b-2)-1)) + b + (b-1)(b^(b-2)-1) is a term. (End)
From Chai Wah Wu, May 18 2021: (Start)
Either c>=3 or y>=3. If c=y=2, we get b^2+b+2=x^2+x-2, i.e. (x-b)(x+b+1) = 4. Since x>1 and b>1, x+b+1>4, a contradiction.
This allows for a faster search algorithm by assuming c>=3 and y>=3. The cases c=2 and y>=3 can be dealt with by picking y>=3 and solving for b in the quadratic equation b^2+b+2=x^y+x-y. Similarly for c>=3 and y=2. This approach was used to confirm a(9). (End)
We have max(c,y) >= 4. Cases (c,y) = (3,2) and (2,3) correspond to elliptic curves with no integral points, while case (c,y) = (3,3) implies (x-b)|6 and leads to quadratic equations with no integer roots. Furthermore, (c,y) = (4,2) corresponds to a quartic curve and has the only solution (b,x) = (3,9); similarly (c,y) = (2,4) has the only solution (b,x) = (3,2); and (c,y) = (4,4) implies (x-b)|8 and leads to cubic equations with no integer roots. Hence, a(n) for n >= 3 correspond to either max(c,y) >= 5 or (c,y) in {(4,3), (3,4)}. - Max Alekseyev, Jan 30 2025
a(11) > 10^32. - Max Alekseyev, Feb 07 2025
EXAMPLE
a(1) = 14 = 3^2 + 3 + 2 = 2^4 + 2 - 4.
a(2) = 88 = 3^4 + 3 + 4 = 9^2 + 9 - 2.
a(3) = 65548 = 4^8 + 4 + 8 = 16^4 + 16 - 4.
a(4) = 33554459 = 2^25 + 2 + 25 = 32^5 + 32 - 5.
a(5) = 387420510 = 3^18 + 3 + 18 = 27^6 + 27 - 6.
a(6) = 1099511627800 = 4^20 + 4 + 20 = 32^8 + 32 - 8.
a(7) = 35184372088855 = 8^15 + 8 + 15 = 32^9 + 32 - 9.
a(8) = 3656158440063002 = 6^20 + 6 + 20 = 36^10 + 36 - 10.
From Michael S. Branicky, May 15 2021: (Start)
The following are terms:
459986536544739960976836 = 7^28 + 7 + 28 = 49^14 + 49 - 14,
1237940039285380274899124273 = 4^45 + 4 + 45 = 64^15 + 64 - 15,
6362685...0216378 (46 digits) = 9^48 + 9 + 48 = 81^24 + 81 - 24, and
1000000...0000070 (61 digits) = 10^60 + 10 + 60 = 100^30 + 100 - 30. (End)
PROG
(Python)
TOP = 100000000
a = [0]*TOP
for y in range(2, TOP//2):
if 2**y+2+y>=TOP: break
for x in range(2, TOP//2):
k = x**y+x+y
if k>=TOP: break
a[k]=1
for y in range(2, TOP//2):
if 2**y+2-y>=TOP: break
for x in range(2, TOP//2):
k = x**y+x-y
if k>=TOP: break
if k>=0: a[k]|=2
print([n for n in range(TOP) if a[n]==3])
CROSSREFS
KEYWORD
nonn,more,changed
AUTHOR
Alex Ratushnyak, Feb 24 2015
EXTENSIONS
a(5)-a(8) from Lars Blomberg, May 19 2015
a(9) from Michael S. Branicky confirmed by Chai Wah Wu, May 18 2021
a(10) from Michael S. Branicky confirmed by Max Alekseyev, Jan 30 2025
STATUS
approved