# Atomic Number Equation Based on Larson's Triplets

December 31, 1980

Dear Mr. Larson

I trust this letter finds you in good health.

After many a summer, I have taken up my ‘pen’ again, to write to you. I decided years ago, that until I has something better top do than just ask questions, I would not bother you directly. So, as you know, I have been writing to N.S.A. members, in particular Frank Meyer.

I have been doing some of my own in-depth study of R.S. and have appeared to come up with something of interest to R.S. IN GENERAL and you in particular. Please make allowance for my lack of COMPLETE understanding, which is a goal to work towards, but difficult to achieve.

As you know, my interest in anything at all is mainly in mathematical aspects, where possible. So, since you started the ball rolling with number triplets, where better for me to start also, and hopefully make my own discoveries, where possible? This renewed vigour, on my part, has come about because I have been invited to submit a paper on R.S. to the science editor of a journal, Helix. Because I have no guarantee that it will be accepted, I felt that I had to do as complete an introduction as I was capable, both philosophically and mathematically.

I have come across problems, some of which I have not solved, some solved, and yet others with theoretical developments to be presented to you for an opinion etc..

1) The algebraic expression for atomic number, I have found to be

ATOMIC NUMBER EQUATION BASED ON LARSON’S TRIPLETS (a,b,c,)

If a = b then this reduces to

If a = b + 1 then it reduces to

 a=b a = b+1 Range of c Z Range of Z a b a b 2 1 -1 to 4 c + 2 1 to 6 2 2 -4 to 4 c + 10 6 to 14 3 2 -4 to 9 c + 18 14 to 27 3 3 -8 to 9 c + 36 28 to 45 4 3 -8 to 16 c + 54 46 to 70 4 4 -15 to 16 c + 86 71 to 102 5 4 -15 to -1 c + 118 103 to 117

2) The a,b,c represent displacements, as originally defined in your books.

a) However, note well, that the sum of 3 algebraic expressions in a,b,c resp. results in (Z+2) (see later) or maybe, yet to be investigated, since only a,b are included in cubic expressions. We should especially note well (Z+2-c)

b) a,b,c have been used later, in a Reciprocity journal, representing the magnetic and electric rotations, confusing to the readers of the books and the journal.

3) This brought me to the full realisation, that I could not come to grips with the problem, “How does anyone, but yourself, know with certainty, the algorithm to determine the rotations for all elements?”

Frank Meyer wrote in Reciprocity a few years ago, that he, his son and Ronald Satz were working on a basic computer-program to not merely work out these rotations, but go further and work out crystal structures.

I have made a chart of all elements with their respective displacements and rotations, but I could not detect any mathematical pattern, furthermore the elements with more than one triplet of rotations were not obviously so to me, as to why they were the ones as distinct from any others, despite rereading the book. I would appreciate such an algorithm from you.

4) I believe that I understand the concept of Ionization level. I see it as an indication of the age of the element in question. I presume that at creation, the elements have atomic mass A = 2.Z

The gravitational charge is, from one point of view, understandable, since I see it as the ‘make-up’ to bring the element in question to the atomic mass, as it would be measured today, this value being determined by its admixture od primary and secondary isotopes, hence its non-integer value. This, of course, does not apply to those rarer elements, whose stated atomic mass is an integer, because, presumably, there is no naturally occurring isotope, so that an admixture is out of the question, and as a consequence the formula A = 2.Z + G does not apply.

I see that the Ionization Level increases with time due to bombardment, followed by some absorption of neutrinos into the lower isotopes in particular.

However, this does not explain tha rarity of some lower isotopes, where the most abundant isotope is not at the bottom of the range.

So I see how the neutrinos build up the lower isotopes atom by atom into the higher isotopes. If there is a higher probability for the neutrinos to be absorbed by the higher isotopes in some cases, and the lower isotopes in other cases, then this needs a good theoretical basis, without any appearance of assumption.

This build up is the increase of Ionization Level, I believe. However I now see an apparent contradiction, since I think that Ionization Level, by definition, can only start at zero, and rise by integral values, in units. Your statement that the terrestrial level for I' being just over unity is an averaged situation. Then I ask:-

A) Why then are some elements at a very high Ionization level today?

Is it because of probability considerations, in which case I feel the need to be shown these considerations, not only for myself, but for others?

B) Why are some elements largely negative, and some just negative, when negativity for Ionization Level is, by definition, impossible?

This brings the definition of Gravitational Charge, ‘G’, into doubt and that it needs alteration to avoid these contradictions (See charts later)

5) My answer is that there is something lacking in the formula, and I have taken the liberty to alter, as a proposal for serious consideration and evaluation, so see whether it can be considered as a logically correct deduction from the original premises, as much as the other formula for G'.

IONIZATION LEVELS IN THE RECIPROCAL SYSTEM OF THEORY

It is stated in the texts how to calculate atomic mass from atomic number A = 2.Z + G where G represents gravitational charge and can be represented by

where the inter-regional ratio 156.4 = 1408/9 and where k = 0

The only alteration, (at this stage, anyhow), being k' in the Z term under the square, since:-

A) k', I believe, has a meaning within R.S.; I am sure that you could explain it in detail; I have only an intuition about it.

B) I stressed earlier that (Z+2) = f(a,b,c) OR (Z+2-c) = g(a,b) and therefore I suspected that k' would turn out to be equal to 2 in many instances, in fact in about 33% more cases than k = 0, as you will see in the accompanying charts.

C) The artifice of using k' is also mathematical, since it is another way to increase, or decrease, the numerator, thus in one foul swoop explaining away an apparent negative I' value in one instance, and avoiding a non-integral I' value in another instance.

6) At last to the charts:-

The column headings are fairly explicit, however to elaborate:-

The first two columns after Atomic Mass (Nos. 3 & 4) show the apparent I-values, derived from your equation for theoretical Atomic Mass, where actual Atomic Mass is used, and k' is taken as 0 & 2 resp. and the equation is solved for ‘I’.

In the next 8 columns, I = 1 throughout and I compare your theoretical Atomic Mass with actual Atomic Mass, on a percentage basis, taking a succession of k-values from -3 thru 4.

The results from comparing firstly those first two columns with each other, then the next 8 columns with each other are both identical and interesting.

But firstly, I opine that out of 103 elements with stated atomic masses, we have to subtract 2 groups of them, in order to find proper meaning for worthwhile deductions.

The first group are those elements, only known from the table to have an integral mass, despite their many unstable isotopes, and therefore cannot be expected to conform to your equation.

Atomic Nos.:-84,85,86,87,89,94,95,96,97,98,99,100,101,102 & 103 (Total=15)

The second group are those elements, whose non-conformity with your equation is so obvious, that it is unreasonable to include them, since we are only comparing those elements, which conform to that equation:

Atomic Nos.:-1,2,3,4,5,6,7,8,10,12,14,15,16,18,19,20 & 28 (Total=17)

RESULTS

Where the percentage figure in the columns is close to 100, they fall into two main groups.

k = 0 :- 13,17,21,26,27,29,30,41,45,47,59,61,63,69,71,

73,74,75,77,78,79,80,81,82,83,88,90,91,92,93.

Total = 30

k = 2 :- 9,11,22,23,24,25,31,32,33,34,35,36,37,38,39,40,42,43,44,

46,48,49,50,51,52,53,54,55,56,57,58,60,62,64,65,66,67,68,70,72,76

Total = 41

So, as you can see, this analysis finds that k = 2 is the hot favorite in the ratio 41:30 i.e. 57.75% of the considered selection, which represents 36.67% more elements favor k = 2

7) Finally, there has to be a precise reason given for non-conformity to your equation. We see, overall, a parabolic type of curve, when plotting Z' against `A', but not the same parabola fits them all.

k = 0 goes well for atomic numbers in the 70's and 80's

k = 2 goes well for atomic numbers in the 30's, 40's, 50's & 60's

However the first 20 are the worst drop-outs, and I feel intuitively that the explanation lies, mathematically speaking, in the function (Z+2-c) or perhaps (Z+k-c), as suggested earlier.

8) Also I am not clear, when you discuss c-matter arriving in this sector, and taking variously 1 or 2 Gravitational Charges, that you don't appear, likewise, to allow for more than one charge in your equation for Atomic mass, as used above.

My regards to your wife and yourself,

Sincerely

David Halprin

This equation covers columns 3 & 4

These 2 equations cover columns 5 to 12

 ATOMIC NUMBER ATOMIC MASS APPARENT IONIZATION   LEVEL ASSUMING   AL = A PERCENTAGE ACCURACY OF LARSON ATOMIC MASS (AL) COMPARED WITH ACTUAL ATOMIC MASS (A) ASSUMING ALL IONIZATION AT LEVEL I = 1 1 2 3 4 5 6 7 8 9 10 11 12 Z A k = 0 k = 2 k=-3 k=-2 k=-1 k=0 k=1 k=2 k=3 k=4 1: HYDROGEN 1.008 -155.19 -17.24 200.95 199.04 198.41 199.04 200.95 204.12 208.55 214.26 2: HELIUM 4.0026 0.1017 0.0254 100.09 99.94 100.09 100.57 101.37 102.49 103.93 105.68 3: LITHIUM 6.941 16.357 5.8886 86.44 86.53 86.81 87.27 87.91 88.75 89.75 90.96 4: BERYLLIUM 9.01504 8.9248 4.4110 88.81 89.02 89.37 89.87 90.51 91.29 92.21 93.27 5: BORON 10.81 5.0688 2.5861 92.74 93.04 93.45 93.98 94.63 95.40 96.29 97.30 6: CARBON 12.0111 0.0482 0.0271 100.39 100.76 101.24 101.82 102.52 103.31 104.22 105.23 7: NITROGEN 14.0067 0.0214 0.0164 100.68 101.09 101.59 102.19 102.87 103.64 104.52 105.47 8: OXYGEN 15.9994 -0.0014 -0.00094 101.00 101.44 101.96 102.56 103.24 104.00 104.84 105.76 9: FLUORINE 18.9984 1.9283 1.2909 95.96 96.39 96.90 97.47 98.11 98.81 99.59 100.43 10: NEON 20.182 0.2847 0.1977 100.65 101.12 101.66 102.26 102.93 103.66 104.45 105.31 11: SODIUM 22.9898 1.2797 0.9163 97.47 97.95 98.47 99.06 99.70 100.39 101.14 101.95 12: MAGNESIUM 24.312 0.3390 0.2490 100.85 101.35 101.90 102.50 103.16 103.87 104.63 105.45 13: ALUMINIUM 26.98 0.9072 0.6814 98.74 99.23 99.77 100.37 101.01 101.70 102.43 103.21 14: SILICON 28.086 0.0686 0.0526 102.45 102.97 103.54 104.15 104.81 105.52 106.27 107.07 15: PHOSPHORUS 30.9738 0.6771 0.5271 99.83 100.34 100.90 101.50 102.14 102.82 103.54 104.31 16: SULPHUR 32.064 0.0391 0.0309 103.17 103.71 104.29 104.90 105.56 106.26 107.00 107.77 17: CHLORINE 35.453 0.8353 0.6687 99.43 99.96 100.52 101.11 101.74 102.41 103.11 103.85 18: ARGON 39.958 1.9063 1.5441 93.72 94.21 94.74 95.30 95.89 96.51 97.17 97.86 19: POTASSIUM 39.10 0.4767 0.3902 101.37 101.91 102.48 103.09 103.72 104.40 105.10 105.83 20: CALCIUM 40.08 0.0313 0.0259 102.56 103.11 103.90 104.31 104.95 105.62 106.33 107.06 21: SCANDIUM 44.956 1.0486 0.8742 98.03 98.56 99.11 99.70 100.31 100.95 101.61 102.31 22: TITANIUM 47.90 1.2606 1.0593 96.68 97.20 97.74 98.32 98.92 99.54 100.20 100.89 23: VANADIUM 50.941 1.4612 1.2368 95.32 95.83 96.37 96.94 97.52 98.14 98.78 99.45 24: CHROMIUM 51.996 1.0853 0.9248 97.74 98.26 98.82 99.40 100.00 100.63 101.28 101.95 25: MANGANESE 54.9380 1.2360 1.0597 96.64 97.17 97.71 98.28 98.88 99.49 100.13 100.80 26: IRON 55.847 0.8903 0.7677 99.17 99.70 100.27 100.85 101.46 102.08 102.74 103.41 27: COBALT 58.9332 1.0587 0.9177 97.88 98.41 98.96 99.53 100.13 100.75 101.39 102.05 28: NICKEL 58.71 0.5408 0.4711 102.19 102.74 103.32 103.92 104.54 105.18 105.85 106.53 29: COPPER 63.545 1.0315 0.9027 98.07 98.61 99.16 99.73 100.33 100.94 101.57 102.23 30: ZINC 65.38 0.9352 0.8219 98.90 99.44 99.99 100.57 101.17 101.78 102.42 103.07 31: GALLIUM 69.72 1.2568 1.1090 96.11 96.64 97.18 97.74 98.32 98.91 99.53 100.16 32: GERMANIUM 72.59 1.3124 1.1625 95.57 96.09 96.63 97.18 97.75 98.35 98.95 99.58 33: ARSENIC 74.9216 1.2817 1.1394 93.77 96.29 96.83 97.38 97.95 98.54 99.14 99.77 34: SELENIUM 78.96 1.4832 1.3230 95.90 94.41 94.94 95.48 96.04 96.61 97.20 97.81 35: BROMINE 79.903 1.2647 1.1317 95.80 96.32 96.85 97.41 97.97 98.55 99.16 99.77 36: KRYPTON 83.80 1.4244 1.2784 94.23 94.74 95.26 95.80 96.36 96.93 97.52 98.12 37: RUBIDIUM 85.466 1.3103 1.1794 95.23 95.75 96.28 96.82 97.38 97.96 98.55 99.16 38: STRONTIUM 87.62 1.2589 1.1362 95.67 96.19 96.73 97.27 97.83 98.41 99.00 99.61 39: YTTRIUM 88.905 1.1216 1.0149 97.05 97.57 98.12 98.67 99.24 99.82 100.42 101.03 40: ZIRCONIUM 91.22 1.0971 0.9951 97.29 97.82 98.36 98.91 99.47 100.06 100.66 101.27 41: NIOBIUM 92.9066 1.0150 0.9228 98.20 98.73 99.27 99.82 100.40 100.98 101.58 102.19 42: MOLYBDENUM 95.94 1.0589 0.9648 97.69 98.21 98.75 98.31 99.31 100.48 101.04 101.65 43: TECHNETIUM 99 1.0999 1.0043 97.20 97.72 98.26 98.81 99.37 99.94 100.53 101.13 44: RUTHENIUM 101.1 1.0586 0.9685 97.67 98.20 98.73 99.28 99.84 100.42 101.01 101.61 45: RHODIUM 102.906 0.9971 0.9140 98.42 98.94 99.48 100.04 100.60 101.18 101.77 102.37

 ATOMIC NUMBER ATOMIC MASS APPARENT IONIZATION   LEVEL ASSUMING   AL = A PERCENTAGE ACCURACY OF LARSON ATOMIC MASS (AL) COMPARED WITH ACTUAL ATOMIC MASS (A) ASSUMING ALL IONIZATION AT LEVEL I = 1 1 2 3 4 5 6 7 8 9 10 11 12 Z A k = 0 k = 2 k=-3 k=-2 k=-1 k=0 k=1 k=2 k=3 k=4 46: PALLADIUM 106.4 1.0647 0.9778 97.57 98.10 98.63 99.18 99.73 100.30 100.89 101.49 47: SILVER 107.870 0.9823 0.9037 98.61 99.14 99.68 100.23 100.79 101.37 101.96 102.55 48: CADMIUM 112.40 1.1136 1.0263 96.93 97.44 97.97 98.51 99.06 99.62 100.20 100.78 49: INDIUM 114.82 1.0960 1.0117 97.13 97.65 98.18 98.72 99.27 99.83 100.40 100.99 50: TIN 118.69 1.1696 1.0813 96.15 96.66 97.18 97.72 98.26 98.82 99.38 99.96 51: ANTIMONY 121.75 1.1879 1.1000 95.87 96.38 96.90 97.43 97.97 98.52 99.08 99.66 52: TELLURIUM 127.6 1.3654 1.2661 93.53 94.03 94.53 95.05 95.57 96.11 96.66 97.21 53: IODINE 126.9044 1.1642 1.0811 96.12 96.63 97.15 97.67 98.22 98.76 99.32 99.89 54: XENON 131.30 1.2501 1.1624 94.92 95.42 95.93 96.45 96.98 97.52 98.07 98.63 55: CESIUM 132.906 1.1846 1.1030 95.77 96.27 96.79 97.31 97.85 98.39 98.94 99.51 56: BARIUM 137.34 1.2641 1.1784 94.62 95.12 95.63 96.14 96.67 97.20 97.75 98.30 57: LANTHANUM 138.91 1.1995 1.1195 95.49 95.99 96.50 97.02 97.55 98.09 98.63 99.19 58: CERIUM 140.12 1.1217 1.0482 96.59 97.09 97.61 98.13 98.67 99.21 99.76 100.32 59: PRASEODYMIUM 140.91 1.0296 0.9632 97.97 98.48 99.00 99.53 100.07 100.62 101.18 101.75 60: NEODYMIUM 144.24 1.0534 0.9865 97.59 98.10 98.62 99.14 99.68 100.23 100.78 101.34 61: PROMETHIUM 145 0.9670 0.9066 98.97 99.48 100.01 101.54 101.03 101.63 102.19 102.76 62: SAMARIUM 150.35 1.0724 1.0064 97.27 97.78 98.29 98.82 99.35 99.89 100.44 100.99 63: EUROPIUM 151.96 1.0233 0.9613 98.06 98.57 99.09 99.61 100.15 100.69 101.24 101.80 64: GADOLINIUM 157.25 1.1162 1.0496 96.52 97.02 97.53 98.05 98.57 99.10 99.65 100.20 65: TERBIUM 158.92 1.0709 1.0079 97.26 97.77 98.28 98.80 99.32 99.86 100.40 100.95 66: DYSPROSIUM 162.50 1.0954 1.0319 96.84 97.34 97.85 98.36 98.88 99.40 99.95 100.50 67: HOLMIUM 164.93 1.0779 1.0163 97.12 97.62 98.13 98.64 99.17 99.70 100.24 100.78 68: ERBIUM 167.26 1.0576 0.9980 97.46 97.96 98.47 98.98 99.51 100.04 100.57 101.12 69: THULIUM 168.934 1.0165 0.9600 98.17 98.67 99.18 99.70 100.22 100.76 101.30 101.85 70: YTTERBIUM 173.04 1.0549 0.9971 97.49 97.99 98.49 99.00 99.55 100.05 100.59 101.13 71: LUTETIUM 174.97 1.0232 0.9679 98.05 98.55 99.06 99.57 100.10 100.62 101.16 101.71 72: HAFNIUM 178.49 1.0409 0.9853 97.73 98.22 98.73 99.24 99.76 100.29 100.82 101.36 73: TANTALUM 180.948 1.0260 0.9720 98.00 98.49 99.00 99.51 100.03 100.56 101.09 101.63 74: TUNGSTEN 183.85 1.0242 0.9710 98.03 98.52 99.03 99.54 100.06 100.58 101.11 101.65 75: RHENIUM 186.2 1.0068 0.9552 98.35 98.85 99.36 99.86 100.38 100.91 101.44 101.98 76: OSMIUM 190.2 1.0347 0.9823 97.83 98.32 98.82 99.33 99.84 100.36 100.89 101.42 77: IRIDIUM 192.22 1.0085 0.9581 98.33 98.82 99.32 99.83 100.35 100.87 101.40 101.93 78: PLATINUM 195.09 1.0051 0.9555 98.39 98.89 99.39 99.90 100.41 100.93 101.46 101.99 79: GOLD 196.967 0.9768 0.9292 98.96 99.48 99.96 100.47 100.99 101.51 102.04 102.57 80: MERCURY 200.59 0.9922 0.9444 98.66 99.15 99.65 100.16 100.67 101.19 101.72 102.25 81: THALLIUM 204.39 1.0108 0.9626 98.29 98.78 99.28 99.78 100.29 100.80 101.33 101.86 82: LEAD 207.19 1.0049 0.9576 98.41 98.90 99.40 99.90 100.41 100.92 101.44 101.97 83: BISMUTH 208.980 0.9760 0.9307 98.41 98.90 99.40 99.90 100.41 100.92 101.44 101.97 84: POLONIUM 210 0.9312 0.8884 99.97 100.47 100.97 101.48 101.99 102.51 103.04 103.57 85: ASTATINE 210 0.8661 0.8268 101.42 101.92 102.42 102.94 103.46 103.99 104.52 105.06 86: RADON 222 1.0576 1.0101 97.31 97.79 98.28 98.77 99.27 99.77 100.28 100.80 87: FRANCIUM 223 1.0128 0.9678 98.25 98.74 99.23 99.72 100.22 100.73 101.24 101.76 88: RADIUM 226.026 1.0106 0.9662 98.30 98.78 99.27 99.77 100.27 100.77 101.29 101.80 89: ACTINIUM 227.03 0.9678 0.9257 99.20 99.70 100.22 100.72 101.22 101.73 102.24 102.76 90: THORIUM 232.038 1.0051 0.9618 98.42 98.91 99.39 99.89 100.39 100.89 101.40 101.91

 ATOMIC NUMBER ATOMIC MASS APPARENT IONIZATION   LEVEL ASSUMING   AL = A PERCENTAGE ACCURACY OF LARSON ATOMIC MASS (AL) COMPARED WITH ACTUAL ATOMIC MASS (A) ASSUMING ALL IONIZATION AT LEVEL I = 1 1 2 3 4 5 6 7 8 9 10 11 12 Z A k = 0 k = 2 k=-3 k=-2 k=-1 k=0 k=1 k=2 k=3 k=4 91: PROTACTINIUM 231.036 0.9264 0.8870 100.20 100.69 101.18 101.68 102.19 102.70 103.22 103.74 92: URANIUM 238.04 0.9988 0.9568 98.57 99.05 99.53 100.03 100.52 101.02 101.53 102.05 93: NEPTUNIUM 237.04 0.9234 0.8849 100.31 100.79 101.29 101.79 102.29 102.80 103.32 103.84 94: PLUTONIUM 244 0.9915 0.9506 98.74 99.22 99.71 100.20 100.69 101.19 101.70 102.20 95: AMERICIUM 243 0.9187 0.8812 100.45 100.94 101.43 101.92 102.43 102.93 103.45 103.97 96: CURIUM 247 0.9336 0.8959 100.12 100.60 101.09 101.50 102.08 102.58 103.10 103.61 97: BERKELIUM 247 0.8812 0.8460 101.41 101.90 102.39 102.89 103.39 103.90 104.42 104.92 98: CALIFORNIUM 251 0.8959 0.8604 101.07 101.56 102.05 102.55 103.05 103.55 104.07 104.58 99: EINSTEINIUM 254 0.8939 0.8588 101.14 101.63 102.12 102.62 103.12 103.62 104.13 104.65 100: FERMIUM 257 0.8917 0.8571 101.22 101.71 102.20 102.69 103.19 103.69 104.21 104.72 101: MENDELEVIUM 256 0.8282 0.7963 102.88 103.38 103.88 104.38 104.88 105.40 105.91 106.43 102: NOBELIUM 254 0.7518 0.7232 104.98 105.48 105.98 106.50 107.01 107.53 108.06 108.59 103: LAWRENCIUM 257 0.7520 0.7237 105.03 105.52 106.03 106.54 107.05 107.57 108.10 108.63 104: 105: 106: 107: 108: 109: 110: 111: 112: 113: 114: 115: 116: 117: LARSONIUM est.???

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