13238717

kp=13238717

#kp is the nearest prime to original Kaelygon magenta #c904bd by taking euclidean distance in RGB space

Closest magenta primes in other colorspaces:
CIE76: 0xC904BD = 13173949
RGB24: CA01BD = 13238717 prime
RGB15: 0x6037 = 29^3+3^5-1
RGB12: 0xD0C = 3340
RGB24 truncated: 0xC0B = 3083 prime


##Extra about 3083
3083^3 = 395^3 + 1012^3 + 3044^3
3083**6948-90 is 24242 digit long prime

#Short facts
##Arbitrary forms
2520*k+1157: 2520*5253+1157=13238717 PRIME
Insert operators: 1+3238*717 = 2321647 PRIME

##Sums
Sum of factorials: 3*10!+6*9!+4*8!+2*7!+5*6!+3*4!+2*2!+1! = 13238717
Sum of two squares: 6619359^2-6619358^2, 181^2+3634^2
Sum of three squares: 3597^2+548^2+2^2
Sum of four cubes: 11^3+95^3+179^3+188^3
Sum of 4-power polynomials: 2*20^4 + 27*25^4 + 2*33^4=13238717

##Misc
First occurance in pi 50298881st digit
Pythagorean triple: 1315508^2+13173195^2=13238717^2
Gaussian prime: 181+3634i which norm is 13238717
Sexy prime pair: (13238717, 13238723)
Non trivial lattice points that a circle with radius 13238717 cross: (1315508,13173195)
Zeckendorf representation: 13238717 = 9227465+3524578+317811+121393+46368+987+89+21+5+0
Palindrome in base 2137: 2*2137^2 + 1921*2137^1 + 2*2137^0 = 2*2137^2 + 1921*2137 + 2

##kp donts
Can not be written as sum of two cubes (exhaustive test range -2200 to 2200 )
Can not be writte as sums of 3 cubes. kp%9=5
Not a sum of 3 consequtive primes
Not a sum of first n primes: Closest sum of primes 2 to 15581 = 13243813 (not prime)
Not delicate


#Neighbor primes:
{..13238689, 13238717, 13238723..}
13238717^2+4 is prime
13238717^2-6 is prime


#Positional primes:
13238717 is 863766th prime
13238717th prime is 241486247


#Delicate primes
kp is not delicate
13238717
prime	 	digit changed
none 	 	8.
10238717 	7.
13738717 	6.
13228717 	5.
13231717 	4.
13238917 	3.
13238747 	2.
none 	 	1.


#Sum of 3 cubes
Cube of kp can be written as sum of 3 positive cubes in 6 distinctive ways:
a^3 + b^3 + c^3 = kp^3 [ 0 < a,b,c = KP ] https://github.com/Kaelygon/cubesum-cuda
403267^3  + 7784369^3 + 12272681^3
5179429^3 + 7227074^3 + 12173060^3
5391033^3 + 7777445^3 + 11918751^3
5633084^3 + 9374269^3 + 10963340^3
5745661^3 + 8854868^3 + 11282720^3
7178816^3 + 8458180^3 + 11038973^3 = 13238717^3


#Kolakoski sequence
Sum of first 8825809 digits of Kolakoski sequence
A054353[8825809]
Kolakoski sequence partial sum primes 5, 7, 19, 23, 29, 37, 41, 43, 47, 59, 61, 71, 73...


#Brussels choice
halve or double any substring in base 10
13238717
+13138717
+1369717
+1369734
+1369768
+1184888
+192888
+96448
+48224
+24112
+2416
+248
+124
+112
+16
=32036129


#Collatz 3n+1
Steps to stop: 148
Highest number reached 241548136


#Compact symbolic notation
kp likely can't be represented with fewer symbols using only "0-9 * + - ^"
Shortest found notations 10 characters when superscript indicator '^' doesn't count as a symbol.
34*624^2-67
29*77^3-740
808*4^7+445
404*8^5+445
202*4^8+445
3639^2-3604
3638^2+3673 # both prime
3634^2+8^5-7
17*92^3+1021
808*4^7+89*5
2^17*101+5*89
113*7^6-927*60


#Large formulated primes, form a*b^n+c where b=kp:

Twin prime 5*13238717^4-6 --- 5*13238717^4-4

13238717^3083+4688
13238717^3083+66716 #Largest known kp base prime

Checked up to n=1536
| 3 | 1*13238717^n+-2 | 12, 26, 144 |

Checked up to n=4788
| 5 | 9*13238717^n+2 | 2,6,54,67,1947
| 5 | 9*13238717^n-2 | 1,62,136,1082,1088

Checked up to n=1000
| 5 | 4*13238717^n+-57 | 2, 5, 6, 12, 168 |
| 5 | 7*13238717^n+-36 | 4, 34, 38, 52, 210 |
| 3 | 2*13238717^n+-1 | 4, 42, 642 |

Checked up to n=7032
| 5 | 1*13238717^n+-24 | 91, 189, 383, 449, 941 |
| 4 | 7*13238717^n+-6 | 2, 10, 290, 516 |
| 3 | 1*13238717^n+-6 | 2, 14, 42 |

Checked up to n=7580
| 5 | 1*13238717^b+-90 | 1, 2, 10, 148, 873 |
| 6 | 1*13238717^b+90 | 1, 5, 8, 116, 2009, 2234 |


#Divisibility rule
P*N = a*10^b+c
P=13238717 N=3 a=4 b=7 c=-283849 P*N=39716151
Test integer num divisibility by P by iterating
	A=num//10^b
	B=num%10^b
	num=B*a-A*c


#abc conjecture formulation
Maximum quality a,b when c=13238717 (exhaustive brute search)
a+b=c
10287648 + 2951069 = 13238717
(2^5 * 3^8 * 7^2) + (11^2 * 29^3) = 13238717

unique factors= 2 * 3 * 7 * 11 * 29 * 13238717
radical= 177372330366
quality= 0.63311566848149
therefore radical is always bigger than c=13238717 for positive a,b