An experiment — factoring a pentatrigesimal “century”

Today, following up from work a few years ago on pentatrigesimal numbers — a number system using all numbers and letters except O [because “O” have I have always viewed as a number because of its extreme similarity to zero] — I experimented with a thirteen-decimal-digit pentatrigesimal century with two goals:

1. to see whether an extremely prime-dense group of 100 numbers would influence the prime density of a cluster as large 1,225 numbers
• 1,225 numbers is, of course, the length of a sequence of numbers whose pentatrigesimal representation differs only in the last two digits
2. to see how well I could convert factors calculated in decimal to pentratrigesimal

The 1,225 numbers chosen have decimal representations ranging from 1,277,956,496,425 to 1,277,956,497,649. This is based on the pentatrigesimal representation of the numbers of the decimal century from 1,277,956,497,400 to 1,277,956,497,499, which is the seventh century with seventeen prime numbers, but the first beyond the reach of prime tables like those at Compoasso, Walter Fendt, and Mark Knows Nothing.

Pentatrigesimal Notation and Factors for All Numbers Not Divisible by 2, 3, or 5 Between 1,277,956,496,425 and 1,277,956,497,649:

In the list below:

1. the bold blue numbers are part of the zero-prime decimal century from 1,277,956,497,000 to 1,277,956,497,099
2. the bold red numbers are part of the seventeen-prime century noted above
3. italicised numbers have pentatrigesimal representations ending in 7, E, L, or T. Any number with pentatrigesimal representations ending in those digits are divisible by 7
1. a number is indeed divisible by 7 if and only if its pentatrigesimal representation ends in:
1. 0
2. 7
3. E
4. L, or
5. T

• JV6W5I04 = 9R6 × 21B5CP
• JV6W5I06 = 3Z × AEH × GTBC
• JV6W5I0C = E9 × ED × 5091521
• JV6W5I0G = W × MEWH3YW
• JV6W5I0I = B × 34 × 5I × 7X × G9M
• JV6W5I0M = 22N × 9JW0V4
• JV6W5I0P = DC × 1H3KLS2
• JV6W5I0T = 7 × D × 12 × 1X × 3S4GH
• JV6W5I0Z = N × 1P × 7R × HI × 4LZ
• JV6W5I11 = JI × 10LVMU2
• JV6W5I17 = 7 × 12X × 2LNT2Y
• JV6W5I1B = AN × 1V85CQ2
• JV6W5I1D is prime
• JV6W5I1H is prime
• JV6W5I1J = D × 1IGNXQLN
• JV6W5I1N = 6D × R1 × 46Q6R
• JV6W5I1U = 18 × 3RI × 4AKQG
• JV6W5I1W is prime
• JV6W5I22 is prime
• JV6W5I26 is prime
• JV6W5I28 = H × J × W × 2F1SIR
• JV6W5I2C is prime
• JV6W5I2E = 7 × B × 2D × 3T8U74
• JV6W5I2I = U × 32 × 30G × 2L31
• JV6W5I2P = 2W × KJ × S2 × F5Y
• JV6W5I2R = 23 × 9IAZ42X
• JV6W5I2X = 12 × 16 × 29 × 73PYZ
• JV6W5I31 = B × D × 1I × 16P × 2PC1
• JV6W5I33 = 3M × 5GKMCTP
• JV6W5I37 = 7 × H × 4S × 3YW × ASZ
• JV6W5I39 = 1NN × BUTIY8
• JV6W5I3D is prime
• JV6W5I3J is prime
• JV6W5I3L = 7 × N × 2J × 38 × 7I × 2FR
• JV6W5I3S = D × 3N76 × EKYZ
• JV6W5I3W = 1R × M3 × I24CH
• JV6W5I3Y = 43Z × 4UP × ZV3
• JV6W5I42 is prime
• JV6W5I44 is prime
• JV6W5I48 is prime
• JV6W5I4E = 7 × J × 1C × 2N4 × 17GB
• JV6W5I4G = 2Y × 23P × 3779N
• JV6W5I4M = 1X × 18GZ × 8CAJ
• JV6W5I4R = Q6 × SLMMSG
• JV6W5I4T = 7 × 2UAZFSKP
• JV6W5I4X = B × N × 2TY × Z0PY
• JV6W5I4Z is prime
• JV6W5I53 = PH × 42H × 6Z3C
• JV6W5I59 = D × D × 16 × 3HWXPR
• JV6W5I5B = 49 × 6YG × NGNU
• JV6W5I5H = J × J × LD9 × 35C3
• JV6W5I5L = 7 × 7 × 2N08 × 5BVD
• JV6W5I5N = 6N × 2ZF0631
• JV6W5I5S = W × MEWH3Z2
• JV6W5I5U = U × 4H × 5C16KY
• JV6W5I5Y = 85Z × 2F2Z62
• JV6W5I64 = H × 15WA2DES
• JV6W5I66 = B × 1T6ZMS3R
• JV6W5I6C is prime
• JV6W5I6G = 54 × 3VXM4E4
• JV6W5I6I is prime
• JV6W5I6M = 5P × 3H9GQRY
• JV6W5I6P is prime
• JV6W5I6T = 7 × B × 60J × 1HIK1
• JV6W5I6Z is prime
• JV6W5I71 = 61 × 2ZJ × 13L8P
• JV6W5I77 = 7 × 1ER × 1ZVRUM
• JV6W5I7B = 29X × 8PGLP8
• JV6W5I7D = 1R × 7FY × 1IHJR
• JV6W5I7H = D × U × Y2Z × 1Y9P
• JV6W5I7J = N × W × 1VC × I9RZ
• JV6W5I7N = J × 1FZY × Q3W4
• JV6W5I7U = FG × 19Z5B14
• JV6W5I7W is prime
• JV6W5I82 = B × H × 3X × Y8FXY
• JV6W5I86 = 56 × S26 × 4YV6
• JV6W5I88 = D × 1AM × 160PHD
• JV6W5I8C is prime
• JV6W5I8E = 7 × 7 × 92 × 1JTX5I
• JV6W5I8I is prime
• JV6W5I8P = B × 2J × 4Y × 50ZKC
• JV6W5I8R = J × CIJ × 2X77W
• JV6W5I8X is prime
• JV6W5I91 = H × 15WA2DEY
• JV6W5I93 = 1I × D438106
• JV6W5I97 = 7 × 18 × 8W × 93EAS
• JV6W5I99 = 12 × 2GW × 7JWRC
• JV6W5I9D = 126 × 1VZ × 9X9S
• JV6W5I9J = 1FP9 × DPZG6
• JV6W5I9L = 7 × 2UAZFSLD
• JV6W5I9S = 1C × ESPH8KB
• JV6W5I9W is prime
• JV6W5I9Y = B × 16 × 6W × 171 × 6ID
• JV6W5IA2 = 38 × 65BEX29
• JV6W5IA4 = 37H8 × 66AFI
• JV6W5IA8 = SM × CE9 × 20ZB
• JV6W5IAE = 7 × YNY × 2Y6S9
• JV6W5IAG = D × 1A8 × 16DDWJ
• JV6W5IAM = AH × EJ × 4JJMU
• JV6W5IAR = 5G × 706 × I6I1
• JV6W5IAT = 7 × U × 3EV5D0R
• JV6W5IAX = J × 6U × 193 × 48W4
• JV6W5IAZ = H × 21 × 81 × BC × 7R6
• JV6W5IB3 = W × 1P × 23 × 6D8G4
• JV6W5IB9 is prime
• JV6W5IBB = I469 × 13CV9
• JV6W5IBH = N × RIN × 14VHD
• JV6W5IBL = 7 × 19S × 27MC3J
• JV6W5IBN = 18 × G5V0XZG
• JV6W5IBS is prime
• JV6W5IBU = B × 1E8 × 19XMZN
• JV6W5IBY = D × H × 29 × 1DS84C
• JV6W5IC4 = 1I × 9M × 1CNT8U
• JV6W5IC6 = 1P4 × BRL9BJ
• JV6W5ICC = BQGJ × 1P9Q8
• JV6W5ICG = B × B × U × 1C × 7C × 28B
• JV6W5ICI is prime
• JV6W5ICM = 9U9 × 20NS9I
• JV6W5ICP = D × 1IGNXQMI
• JV6W5ICT = 7 × N × 4B4IY5I
• JV6W5ICZ = 34 × 6D7ZQIR
• JV6W5ID1 is prime
• JV6W5ID7 = 7 × 8JC × BLERN
• JV6W5IDB = 9G × 23HV1MG
• JV6W5IDD = 1VFH × ALT79
• JV6W5IDH = 12 × 76 × 2LPH9G
• JV6W5IDJ = S8 × 4S4 × 5C5H
• JV6W5IDN = 9X × 3682871749
• JV6W5IDU = 49 × 4B × 2047VX4
• JV6W5IDW = H × H × 2E6RWNJ
• JV6W5IE2 is prime
• JV6W5IE6 = D × J × JR × 4ZMAN
• JV6W5IE8 = 2W × 24M3 × 37ZB
• JV6W5IEC = B × 1R × IH × 1YMZW
• JV6W5IEE = 7 × 6D × LM × Q7UC
• JV6W5IEI = Q8 × 5NH × 4V2D
• JV6W5IEP is prime
• JV6W5IER = 32 × C1 × IWNPD
• JV6W5IEX = D × D × 43Z57CI
• JV6W5IF1 = 21 × 9SPKSD1
• JV6W5IF3 is prime
• JV6W5IF7 = 7 × 2UAZFSM6
• JV6W5IF9 = J × 23 × 2S × A9 × LKZ
• JV6W5IFD is prime
• JV6W5IFJ is prime
• JV6W5IFL = 7 × B × 12 × 3AR × 2KDP
• JV6W5IFS = U × 1BS × HXV7U
• JV6W5IFW = 3X × IU × 9F53S
• JV6W5IFY = 9Z × 1B3 × 1HY6P
• JV6W5IG2 = 3Z × 501QQAY
• JV6W5IG4 = 1X × 1T8 × 5R0YU
• JV6W5IG8 = B × 2Y × E1 × 1IKA6
• JV6W5IGE = 7 × D × W × 5A6 × 1M24
• JV6W5IGG = 29 × 7R × 508 × 7Y3
• JV6W5IGM = 2D × AEN × T4T3
• JV6W5IGR = N × V7WSM3S
• JV6W5IGT = 7 × 7 × 7 × H × 461QYI
• JV6W5IGX = 3R × 1Y9 × 2Q8FD
• JV6W5IGZ = 16 × 1G2 × BLTQX
• JV6W5IH3 = 20H × 9V73UU
• JV6W5IH9 = EW × 6PW × 6YJU
• JV6W5IHB = BPR × 1PDEJR
• JV6W5IHH = B × DX × 4IZ1ER
• JV6W5IHL = 7 × 3M × SCY6U4
• JV6W5IHN = 7P × X9 × 2T51Y
• JV6W5IHS = H × 15WA2DFG
• JV6W5IHU = 1C × 1P × 1R × 1R × 2W3N
• JV6W5IHY = 6J × 1171 × 365BX
• JV6W5II4 = B × 1T6ZMS4U
• JV6W5II6 = W × 1I × 39D × 41LP
• JV6W5IIC = 1093 × 1169219119
• JV6W5IIG = 7901 × 161746171
• JV6W5III = J × 56 × 72M7N2
• JV6W5IIM = D × 7I × CKS × JS9
• JV6W5IIP = B4 × KS × 30DW8
• JV6W5IIT = 7 × YX × 2ZW × Z72
• JV6W5IIZ = 6N × 1IN1 × 1Y43
• JV6W5IJ1 = 18 × AP3 × 1HXZJ
• JV6W5IJ7 = 7 × 34 × 2H4 × CTF6
• JV6W5IJB = 16 × GYG4E3W
• JV6W5IJD = B × B × D × N × NIV82
• JV6W5IJH = 1EX × DXGK06
• JV6W5IJJ = QKUU × S5LR
• JV6W5IJN = IGB × 12MMRI
• JV6W5IJU = 12 × 20S × 9A7VW
• JV6W5IJW is prime
• JV6W5IK2 = D6 × 1R2 × V8Y6
• JV6W5IK6 is prime
• JV6W5IK8 = BG × 1U3 × Y4V6
• JV6W5IKC = 3U1 × 56IZMC
• JV6W5IKE = 7 × A3 × 9EC × 11LX
• JV6W5IKI = 1C × 1UU × 7ZHFR
• JV6W5IKP = H × J × N × N × 17J × 43I
• JV6W5IKR = U × 13WP × LJTB
• JV6W5IKX = RRB × QZIXM
• JV6W5IL1 = K9 × 8MW × 3YSZ
• JV6W5IL3 is prime
• JV6W5IL7 = 7 × 1I × 1P × 4N × 8CDR
• JV6W5IL9 = B × 1T6ZMS54
• JV6W5ILD = 2D × 8D5F8N1
• JV6W5ILJ is prime
• JV6W5ILL = 7 × D × 1MZ × 1QU × 2MW
• JV6W5ILS = J × GZJ × 25DLC
• JV6W5ILW = B × 7M4 × 89UAJ
• JV6W5ILY = 12 × J8 × 52G × 6R3
• JV6W5IM2 = IW × 32G × BZMC
• JV6W5IM4 = 2Y × 793 × XIXB
• JV6W5IM8 = IUW1 × 11VJ8
• JV6W5IME = 7 × 7 × U × 6QI × 2J28
• JV6W5IMG = 4B × 4L4UGKB
• JV6W5IMM = H × AN × 2C9 × 1M53
• JV6W5IMR = 2HZ × 7WKKZ9
• JV6W5IMT = 7 × 2L2 × 1362QM
• JV6W5IMX is prime
• JV6W5IMZ = 38 × 65BEX2D
• JV6W5IN3 = D × 1IGNXQNB
• JV6W5IN9 is prime
• JV6W5INB = N × EY × 20SW7Z
• JV6W5INH = W × MEWH3ZM
• JV6W5INL = 7 × H × 23 × 2T15X3
• JV6W5INN = 1FY × DMM316
• JV6W5INS = B × 1X × 4H × 7CKZ8
• JV6W5INU = D × 3IX × F3N7J
• JV6W5INY = J × 18 × 2J × 11H × B88
• JV6W5IP4 = 10C × JNFUQC
• JV6W5IP6 = CU × 1J6PBMZ
• JV6W5IPC = 2TJ × 6ZN × 10BB
• JV6W5IPG = I7C × 1366XD
• JV6W5IPI = 4CW × 4J59SD
• JV6W5IPM = N × V7WSM44
• JV6W5IPP is prime
• JV6W5IPT = 7 × 1R × 2QZ × KSC6
• JV6W5IPZ = H4 × 15LQJ0R
• JV6W5IQ1 = B × J × 3BEQ53Z
• JV6W5IQ7 = 7 × 7 × E6JX8YD
• JV6W5IQB = D × 1IGNXQNH
• JV6W5IQD = A87Z × 1XXAM
• JV6W5IQH = 29 × BP × RCH4X
• JV6W5IQJ = H × W6 × 1E6 × XNX
• JV6W5IQN = B × 4N × H6 × 9QTW
• JV6W5IQU = 8S × 2Q4 × U5RY
• JV6W5IQW = 1C × RJ × 1EI × DSJ
• JV6W5IR2 = D × 1IGNXQNJ
• JV6W5IR6 = 12 × ISLMUU3
• JV6W5IR8 = 5M × 3IHXCPU
• JV6W5IRC = 16 × 2S × 3PZ × 1MN9
• JV6W5IRE = 7 × 18 × LG × 3RV0P
• JV6W5IRI = H × 92 × 92 × PD × Q2
• JV6W5IRP = 2TM9 × 71NXR
• JV6W5IRR is prime
• JV6W5IRX = B × 54 × CCHX7N
• JV6W5IS1 = W × 81 × 2SRQ1R
• JV6W5IS3 is prime
• JV6W5IS7 = 7 × J × 21 × 21 × 19G09
• JV6W5IS9 = N × 1I × JYLMVG
• JV6W5ISD = U × DC × 1N9 × 12S9
• JV6W5ISJ = B × D × KY × 84CNR
• JV6W5ISL = 7 × 1X × AFG × 4YWJ
• JV6W5ISS = 23 × 1A3J × 7DN6
• JV6W5ISW is prime
• JV6W5ISY = 1P × 8D × 1E97MP
• JV6W5IT2 is prime
• JV6W5IT4 is prime
• JV6W5IT8 is prime
• JV6W5ITE = 7 × PJ × 2PN × 1HCW
• JV6W5ITG = H × 4Y × 771 × 157G
• JV6W5ITM is prime
• JV6W5ITR is prime
• JV6W5ITT = 7 × B × W × BZ × UT2G
• JV6W5ITX is prime
• JV6W5ITZ is prime
• JV6W5IU3 = ED × 27Z × LPZP
• JV6W5IU9 is prime
• JV6W5IUB is prime
• JV6W5IUH is prime
• JV6W5IUL = 7 × 2UAZFSP8
• JV6W5IUN is prime
• JV6W5IUS = D × 1V6 × TQ68Z
• JV6W5IUU = 66H × 6AN × HTZ
• JV6W5IUY is prime
• JV6W5IV4 is prime
• JV6W5IV6 is prime
• JV6W5IVC is prime
• JV6W5IVG = J × 8Y × 11N × 3WQG
• JV6W5IVI = D × 4N × G89 × PRD
• JV6W5IVM is prime
• JV6W5IVP = B × U × 22J × 11T14
• JV6W5IVT = 7 × 7 × 83 × 1REG3U
• JV6W5IVZ = 31H × 6IHGE2
• JV6W5IW1 = 16 × 6J × 2KPLX4
• JV6W5IW7 = 7 × N × 4B4IY5M
• JV6W5IWB = B × 18 × 1P × 1AR × NC8
• JV6W5IWD = H × 9M × 1DD × 32JJ
• JV6W5IWH is prime
• JV6W5IWJ = J × 2W × 3X × 1JH × 22U
• JV6W5IWN = 34 × FX × E0XZ1
• JV6W5IWU = 4S × 7652434117
• JV6W5IWW = 2S × 3R × 1X0NTY
• JV6W5IX2 = G3 × GY × 2J9RN
• JV6W5IX6 = 7X × 2HUEXIY
• JV6W5IX8 = 9G × 7DB × 9YL8
• JV6W5IXC = H × U × W × 1KQASZ
• JV6W5IXE = 7 × C72 × 84VRB
• JV6W5IXI = N × 12 × 436 × 6ZMD
• JV6W5IXP is prime
• JV6W5IXR = D × 4H × BX8WKW
• JV6W5IXX = UI × NJECZU
• JV6W5IY1 = 7P × 2KFVN2J
• JV6W5IY3 = 32 × F0R × F4UU
• JV6W5IY7 = 7 × B × B × VNC × XSI
• JV6W5IY9 is prime
• JV6W5IYD = 16 × 3Z1H × 49DJ
• JV6W5IYJ is prime
• JV6W5IYL = 7 × 7 × 5I × 1IZ × 1NDX
• JV6W5IYS = 18 × G5V0XZZ
• JV6W5IYW = 2Y × 1XC × 3HS66
• JV6W5IYY = 1C × 17H × C6GYC
• JV6W5IZ2 is prime
• JV6W5IZ4 = W × 2J × 7ER × 16KR
• JV6W5IZ8 = D × D × 1RR × 2BLIH
• JV6W5IZE = 7 × 1KDN × 1SRE9
• JV6W5IZG = B × 1T6ZMS6B
• JV6W5IZM = 12 × 2KW × 788P6
• JV6W5IZR is prime
• JV6W5IZT = 7 × J × CJ × EKHHJ
• JV6W5IZX is prime
• JV6W5IZZ = D × 15Y × 1AQ0AW

Analysis:

If we count the primes in the “century” [actually, of course, a sequence of one thousand two hundred and twenty-five integers, or twelve and a quarter times as many as a conventional decimal century], we find that there are a total of sixty-three primes.

According to the prime number theorem, there should be an average gap of approximately 27.876 between primes over these numbers. This would mean that there should be about 43.944 primes over the pentatrigesimal “century”, which is approximately nineteen fewer than the number actually observed.

This does imply that the extremely prime-rich decimal century from 1,277,956,497,400 to 1,277,956,497,499 has effects that extend to a full pentatrigesimal “century” more than twelve times as large.