## Thealt.co

Combinatorics and algebra have been used to find equations for the smallest integer with a certain length in an integral base. How-ever, improper fractional bases have not been explored in much depthsince their discovery in the 1930s. In this study, I discovered an orig-inal formula for the smallest integer with a specific digit length in animproper fractional base.
I wrote an original computer program to convert integers from base 10 to any improper fractional base. I used this program to find 100combinations of length, improper fractional base, and the smallestinteger with that length in that fractional base.
combinatorics, and difference equations to attempt to find a methodto predict the smallest integer with a specific length in an improperfractional base.
I then used number theory to evaluate the divisibility requirements of the numbers, and discovered a recursive formula for the smallestinteger with a specific length in a given improper fractional base. Iused this formula to find an equation for the number of integers in animproper fractional base with a certain length. The formula may alsobe useful in encryption with improper fractional bases.
Common Bases . . . . . . . . . . . . . . . . . . . . . . . . . .
Historical Bases . . . . . . . . . . . . . . . . . . . . . .
Computational Bases . . . . . . . . . . . . . . . . . . .
Unusual Bases . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bases 3 and -2 . . . . . . . . . . . . . . . . . . . . . . .
Irrational Bases . . . . . . . . . . . . . . . . . . . . . .
Transcendental Bases . . . . . . . . . . . . . . . . . . .
Improper Fractional Bases . . . . . . . . . . . . . . . .
Smallest Integer Patterns . . . . . . . . . . . . . . . . . . . . .
Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conversion Using Powers of a Base . . . . . . . . . . . . . . .
Converting Fractional Bases . . . . . . . . . . . . . . . . . . .
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Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fractional Base Encryption of Short Messages . . . . . . . . .
Security Issues . . . . . . . . . . . . . . . . . . . . . . . . . . .
10 Checklist for Adult Sponsor / Safety Assessment Form (1) Since the beginning of civilization, systems of numeration have been impor-tant to mankind. People still use basic one-to-one correspondence with ourfingers to find base 10. As society increased in complexity, so did our systemsof numeration. Today we have a wide variety of counting methods which areno longer directly dependent upon our physical features.
Integral (whole number) bases are the most common. Historically, bases 10,20, 60, and 12 have been used most widely; other bases occasionally used are2, 8, 16, and 3.
We usually have 10 fingers and 10 toes, therefore it is convenient to use base 10. The Phoenicians and the Mayans used base 20, presumably countingboth fingers and toes. The Sumerians used base 60 for numbers over 20,which is the origin of our minute and degree measures. 60 is a useful basebecause it has a large number of factors. Of the fractions in the form 1xwith 0 < x ≤ 60, in base 60, 12 have 1-digit decimal representations. Thismakes it much easier to convert between hours and minutes. In base 60, 1hour 5 minutes is equal to 1.05; hours instead of 1.3 hours in base 10 (Base60 numbers are written with a comma or semicolon separating the base-60digits).
Some early civilizations used the 3 joints on their 4 fingers of one hand to arrive at base 12. Similar to 60, 12 has many factors. Of the first 12 naturalnumbers’ reciprocals, only 6 do not have finite decimal representations. Base12 has influenced us widely: for instance, 12 items is a dozen and 12 dozenis a gross.
A computer’s storage is similar to a great number of on-off switches. Theseare also known as 2-state devices since they have 2 possible states, corre-sponding to 1 and 0 in base 2. It is impossible to store more than a 1 or a0 on a single 2-state device, therefore computers convert to base 2 to storeinformation. Base 8 is used when a user-defined number has a power of 8 (but not of 16) as an upper bound. When base 8 is used for such an application, onlya third as many digits are needed to represent the number as there are inbase 2. As a result, base 8 numbers are easier to read compared to base 2.
Base 16 is used for applications when the upper bound is a power of 16; sincefour base 2 digits can be represented in one base 16 digit, the number is morecompact compared to base 2 or 8, even further improving readability. Bases 3 and negative 2 were used in some early computers. Base 3 isregarded by Hayes as the most efficient integral base, since it has the theo-retically smallest ratio of the average length of the numbers to the numberof symbols used as digits of integral bases. Base -2 was attempted in some computers. Simple multiplication and division were more difficult than in base 2, so it was abandoned. At thattime, computers required an extra bit to represent the sign of a number.
Although base -2 had the advantage that no bit was needed to represent thenegative sign, the space savings were negligible and the problems with arith-metic make it less efficient. Present computers can represent the sign ofa number without using an extra bit, negating any space-saving advantageof base -2.
3-state devices are not base 3 in the traditional sense of the word. 3-state devices use what is called “balanced ternary”: instead of having values 0, 1,or 2, a 3-state device can have value of either -1, 0, or 1. Thus computersusing base 3 take only 1 evaluation to determine whether a number is larger,smaller, or equal to another number, yielding values -1, 0, or 1. Although base 3 is more efficient in theory, it has been difficult and expensive to im-plement ; 3-state devices have been too inefficient to compete with base2. Currently, base 2 is more practical since 2-state devices have been op-timized since base 2 was first used for computing, and thus 2-state devicesare very reliable and cheap. However, recent advances in 3-state hard driveshave made base 3 computing much closer than before. An irrational number has an infinitely long, non-repeating decimal portion; itcannot be represented as a ratio of two integers.[11, pp.102-104] For instance,π (pi, the ratio of a circumference of a circle to its diameter) is an irrationalnumber since it cannot be represented as a ratio of two integers. Bergmaninvestigated irrational bases in 1957; he discovered that numbers do notnecessarily have a unique representation in an irrational base.His workhas been important in developing faster algorithms in computer science.Though not common, irrational bases are used occasionally; base φ (the golden ratio, phi, 1+ 5 ) is the most common irrational base. By definition, a transcendental number cannot be a solution of a polynomialwith integral coefficients. All transcendentals are irrational numbers, butnot all irrational numbers are transcendental. π and e (the base of naturallogarithms) are transcendental, but φ is only irrational.  φ is a root ofx2 − x − 1 = 0 while π and e cannot be roots of a polynomial with integralcoefficients.[12, p.114] Since transcendental bases are impractical, they are not often used. Every integer greater than the transcendental base has infinitely many digits in itsrepresentation and only multiples of the transcendental base have finite dec-imal representations. However, it has been shown that base e is theoreticallythe most efficient base out of every possible base. An improper fractional base is rational, expressible as a ratio of two integers,and greater than 1. Improper fractional bases, such as 3 , were discovered by A. J. Kempner in 1936. Besides the method of conversion to and fromfractional bases, little thought had been given to these bases.
One of the main differences between integral bases and improper fractionalbases is in the smallest number with a specific length in a given base. In anintegral base, these numbers follow a exponential progression for digit lengthsgreater than 1. In base 10, the smallest integer with each number of digits ismultiplied by a factor of 10 between n digits and n + 1 digits for n ≥ 2. Thesmallest 1-digit positive integers is 1, 2-digit integer is 10, of 3, 100, and so on.
In an improper fractional base, the numbers are extremely variable even from one base to another similar non-reducable base. For fractional baseintegers there is no obvious pattern. In base 3 , the smallest 1-digit integer is 0, the smallest 2-digit integer is 3, 3-digit integer, 6, 4-digit, 9, 5-digit, 15,and the smallest 6-digit integer is 24.
Combinatoric techniques yield an equation which predicts the smallest integer in the integral base b. In an integral base, any digit can be in anyposition in a number, except 0 cannot be the first digit. Thus in base b, thereare b − 1 1-digit numbers (not including 0), (b − 1)b 2-digit numbers, (b − 1)b23-digit numbers, etc. Let the number of integers with length k in base b beg(l). Then the smallest integer with length l in base b is 1 + g(1) + g(2) +g(3) + · · · + g(l − 1), since this smallest integer will come immediately afterall the integers with length less than l. This may be expressed as: Now there are b − 1 choices of digit for the first digit of a number in base band b choices of digit for each subsequent digit in the number, thus g(l) =(b − 1)bl−1. Thus the smallest number with l digits in base b is equal to Distressingly, this equation does not work for fractional bases.
Is there an equation for the smallest integer with a given number of digits(length) in any fractional base? I sought to find such an equation which usesthe fractional base and the desired length.
A base, also known as a radix, usually indicates how many unique symbolsmay be used to form a number. In base 10, a digit in a number could be 0,1, 2, 3, 4, 5, 6, 7, 8, or 9, for a total of 10 unique symbols. In base 3, a digitcould be either 0, 1, or 2. Fractional bases, expressible in the form p , are somewhat different. In- stead of using digits up to p − 1, in which there would be a fractional number of digits, we use p unique digits: 0, 1, 2, 3, · · · , p − 2, p − 1. For instance,in base 3 , we can use the digits 0, 1, and 2. We generally denote a base b number n with nb, read “n base b”.
There are several methods of converting an integer n to and from base 10.
The most common involves powers of the new base. To convert from base 10to base b using this method, find the largest power of b which is smaller thann and let the exponent be i. The leftmost digit is the largest number which,when multiplied by bi, is smaller than n. We then subtract this product fromn and make this the new n and subtract 1 from i and let this be the new i. Then, until i = 0, we repeatedly find the largest number whose product with bi is smaller than n, and let this number be the next digit to the rightof those which we have found. Next, subtract the product of that numberand bi from n, and subtract 1 from i and make this i. When i = 0, we letthe remaining number be the last (rightmost) digit. In base 10, for instance,the number 37 is equivalent to (3 · 101) + (7 · 100). (Generally x0 = 1, withthe exception of x = 0.) To convert n back to base 10, we count how many digits n has. We sub- tract 1 from this number of digits and let it be i. We then add the productof the first digit and bi, the second digit and bi−1, the third and bi−2, and soon. This number is the base 10 representation of nb. 11013 (1101, base 3) isequal to (1 · 33) + (1 · 32) + (0 · 31) + (1 · 30), which is 37 in base 10.
The above method, while most common, only works for integral bases; when we attempt to use this method for an improper fractional base, thenumber ceases to be integral. For example, to convert 3 (base 10) to base 32 this can be multiplied to get a number smaller than 3 is 1, so the leftmostdigit is 1. The remainder is 3 , and the next smaller power is 3 , so the next digit is 0. The next smaller power is 1, so the third digit is also 0. But nowto the right of the decimal point we must represent 3 : we cannot represent this number without a decimal point using this conversion technique. It islogical that an integer should be represented without any digits to the rightof the decimal point in any rational base, though, so this method cannot beused for improper fractional bases.
Another method for conversion, which fulfills the requirements to produceintegers in fractional bases, follows. Designate the desired base p number . . . d5d4d3d2d1d0 where dk represent the digits. We then let the original num-ber in base 10 be d0. After this, while any of the dk are greater than or equalto p, we subtract p from dk and add q to dk+1. For instance, to convert 19 from base 10 to base 3 , d using dk, we must simply separate the digits in the base b number somehow.
We can use a line to separate the digits, as: We subtract 6 3’s from 19 and add 6 · 2 to the digit immediately to its left: Since 12 > 3 we again subtract 3’s until the digit is smaller than 3. Wesubtract 4 3’s from 12 and add 8 to the digit immediately to the left: and again. 4 > 3, so we subtract a 3 and add a 2, for a final result of or 21201 3 (base 3 ).We generally denote a base b number n with nb, read “n We can use this process in reverse to convert to base 10: we separate the digits, just as before. Then, starting at the left, we multiply each digit by thebase b and add this to the digit immediately right. For instance, to convert21201 from base 3 to base 10 we separate the digits: Multiply the first by 3 and add to the second: Multiply the (new) first digit by 3 and add to the (new) second digit: Finally multiply the first digit by 3 and add to the second digit to get I created an original computer program and converted large numbers of base10 integers to various fractional bases. I used tools from combinatorics, num-ber theory, and polynomial analysis to find an equation appropriate to thedata.
def DecimalToFractionalBaseConversion(DecimalNumber, FractionalBaseNumerator,FractionalBaseDenominator): ArrayOfDigits=[DecimalNumber]while ArrayOfDigits>=FractionalBaseNumerator:ArrayOfDigits=[ArrayOfDigits]+ArrayOfDigitsArrayOfDigits=ArrayOfDigits % FractionalBaseNumeratorArrayOfDigits=ArrayOfDigits-ArrayOfDigitsArrayOfDigits=ArrayOfDigits*FractionalBaseDenominatorArrayOfDigits=ArrayOfDigits/FractionalBaseNumeratorreturn ArrayOfDigits My program above takes as input the base 10 integer, the improper frac- tional bases’ numerator, and the improper fractional bases’ denominator.
The program then creates ArrayOfDigits, which will be used as an array.
Initially, ArrayOfDigits contains one element. Instead of making its boxesat the beginning of the conversion process, as we do when converting to animproper fractional base, ArrayOfDigits adds extra boxes only when needed.
As in our method of conversion, initially the first box of ArrayOfDigits con-tains the base 10 integer.
Then the program checks to see if the first member of ArrayOfDigits, the leftmost box, is bigger than FractionalBaseNumerator, the numerator of thefractional base. If it is, another member or “box” is added to the beginningof ArrayOfDigits.
The program next copies the former first member of ArrayOfDigits to the new first member of ArrayOfDigits. The second member of ArrayOfDigitsis taken modulus FractionalBaseNumerator, which is simply the remainderwhen that member is divided by the numerator of the fractional base. This,the second member, is subtracted from the first member, so now the firstmember is divisible by FractionalBaseNumerator, and the first member ismultiplied by the inverse of the base. This is the same as subtracting the nu-merator and adding the denominator to the next digit for those digits. Theprogram checks again to see if the first member is bigger than the numerator,and repeats if this is true.
When all of the members are smaller than the numerator of the fractional base, the program returns ArrayOfDigits - the contents of our boxes whenwe are done converting.
I used this program to convert the first 5000 positive integers to bases 3 , 4 , 5 , 7, 7, 7, and 8. In each of these bases, I found the smallest integer in that base with various lengths (1 digit, 2 digits, 3 digits, etc.). A fractionalbase, a length, and the smallest integer in that base with exactly that lengthformed 100 data sets.
• I chose base 3 since it has the smallest possible irreducible numerator and denominator for an improper number, thus it is easiest to examineof improper fractional bases.
• I chose bases 4 and 5 since the denominators are only 1 smaller than the numerators. I could easily test an equation for base 3 with a similar base to find the proper generalization for any base with the numerator1 larger than the denominator.
• I chose bases 7 , 7 , and 7 since the denominators were more than 1 smaller than the numerator. It would be possible to generate a equa-tion that would work for the bases with the numerator 1 greater thanthe denominators, but it would be very hard to find an equation that works for both those 3 bases with numerator 7 and the 3 bases withthe numerator 1 greater than the denominator.
• Finally, I chose base 8 since it is a reducable improper fraction. I wanted to test any possible equation on at least one reducable fractionalbase. If a base were not in lowest terms, an equation dependent onthe base being in lowest terms would fail for a reducable improperfractional base. I wanted to insure my equation was not dependent onthis condition.
I initially used difference equations to search for the next member of thesequence of smallest integers with various lengths and fractional bases. How-ever, in all tests the value predicted by the difference equations differed fromthe correct value by a significant amount. Given d + 1 values of any poly-nomial with degree d, all other values of the polynomial may be predicted.
Since the difference equations failed with up to 10 values, evidently the equa-tion is not a polynomial or has degree more than 9.
If the graphs of the smallest integers with n digits in the above bases were similar, I would expect the equation to be a polynomial. Although all graphswere somewhat exponentially curved, the similarities were inconsequential.
The negative results from both graphing and difference analysis indicate thatthe desired equation is not a polynomial. The piecewise nature of the graphssuggested floor or ceiling functions were at work.
Combinatoric techniques have been used in the past to find the equation for the smallest integer with a specific length in an integral base. I hopedsimilar methods could be used to find the desired formula. I tried to find theformula by using base p and then, one digit at a time, converting to base pqin the hope that a pattern would emerge. Sadly, I found no pattern betweenthe conversions to p .
Eventually, I employed number theory to discover the method of forming the smallest number with n digits, by examination of divisibility require-ments. I then tested this recursive formula with all data created by the I discovered some divisibility properties of improper fractional base integers.
With this knowledge, I discovered an original formula for the smallest inte-ger with a specific length in an improper fractional base, which showed myhypothesis to be true. I also discovered an original equation for the numberof integers with a specific length in an improper fractional base, and a newmethod of encryption.
Theorem. For any improper fractional base p , the smallest integer with n digits in its representation in base p is and x is the smallest integer larger than or equal to x.
(To find f a(b), substitute b into f (x). Substitute this result into f (x) again, and substitute that result into f (x) yet again, until we have appliedf (x) a times to b.) Lemma. Let an integer in base p be represented as d the (k − 1)th digit to the right of the rightmost digit. Then all of the integersd0, d0d1, d0d1d2, · · · , d0d1d2 · · · dj−1 are divisible by q.
Proof. Each of the integers d0, d0d1, d0d1d2,· · · , d0d1d2 · · · dj−1 may be repre-sented by d0 · · · dm, where m is a positive integer. According to our method ofconversion from base 10 to a fractional base, each of these numbers d0 · · · dmis created by subtracting p from dm+1 and adding q to dm repeatedly. Thusd0 · · · dm in base p can be represented as qn, where n is the number of sub- tractions of p from dm+1; since n is integral, q | qn, as desired.
Proof of Theorem. The smallest number with exactly n digits will have afirst digit of q, since d0 must be divisible by q. Every subsequent digit dm upto but not including the last digit on the right dn will be the smallest suchthat the number d0d1d2 · · · dm is divisible by q.
If we have already found the smallest possible number d0d1d2 · · · dm−1 and we wish to find the smallest possible integer d0d1d2 · · · dm, we appenda 0 on the end of this number and find the smallest possible dm so thatq | d0d1d2 · · · dm−10 + dm. This is the smallest possible digit dm such thatd0d1d2 · · · dm is divisible by q.
When we append a 0 on the end of the integer d0d1d2 · · · dm−1, we are simply shifting the digits of the integer 1 place to the left, which is equivalentto multiplying the number by our base p . Thus d integer such thatq | (d0d1d2 · · · dm−1) p +d m. This process is equivalent to dividing (d0d1d2 · · · dm−1) p by q, rounding up, and multiplying the result by q, giving the numberd0d1d2 · · · dm. Mathematically, This can be expressed as a function which takes as input d0d1d2 · · · dm−1 If we take the starting digit d0 = q of the smallest number, when we apply f (x) to q, we find the first 2 digits of the number; if we apply it tothis result we get the first 3 digits, and so forth. However, the whole numberd0d1d2 · · · dn does not have to be divisible by q, so we cannot apply f (x) tod0d1d2 · · · dn−1 to find d0d1d2 · · · dn. Instead, the smallest number with n dig-its ends in 0, so to find d0d1d2 · · · dn we multiply d0d1d2 · · · dn−1 by p , our base.
The application of a function f to an input i z times is denoted by f z(i). For instance, f (f (f (5))) = f 3(5). When we apply the function aboven − 2 times, we find the integer d0d1d2 · · · dn−1. To find the full numberd0d1d2 · · · dn, we multiply d0d1d2 · · · dn−1 by p . Thus the smallest number This novel equation was tested with all 100 sets of data and will work for This formula may also be used to find the number of integers with a specificlength. To find the number of integers with a specific length in an improperfractional base, we can take the difference of the smallest integer with thatlength and the smallest integer with one more digit. Thus the number of in-tegers in an improper fractional base with n digits in the improper fractionalbase p is equal to Fractional bases may encrypt data by converting the message to a number,then to a fractional base, and then finally breaking it up and converting itback to a string of letters. My equation is useful to fractional base encryptionwhen a long message must be split into several parts.
Fractional Base Encryption of Short Messages Let us encrypt “Word Power”: A space may be represented as 00, ‘a’ can be01, ‘b’, 02, etc. up to ‘z’ being 26, with lowercase and uppercase letters identi-cal. The numerical representation of this message is thus “23151804001615230518”.
A fractional base with no more than 27 digits is necessary to encrypt this message, since there are 27 possible characters; if we used a base whosenumerator is less than 27, the encrypted message would be longer and wouldhave fewer unique characters. If we used a base whose numerator was morethan 27, it is possible that a digit more than 26 would appear in the numberin that base; we only have 27 characters, so that digit could not be convertedto a character. The base 27 is fairly quick to evaluate, and we find that our in base 27 . Now converting the digits into the corresponding letters, our message ends up as ‘tuiwlxfyadyygjmr’.
To decrypt this encrypted message, convert the encrypted message ‘tui- wlxfyadyygjmr’ back to the base 27 number Then convert this number to base 10 and convert back to letters by our orig-inal method.
If a long message is to be encrypted, it must be divided into several sec-tions to encrypt separately, then combined into one string. After splitting the message into sections, it is necessary to find a base such that all of theencrypted sections have the same length. If the sections differ in length,then the message is undecipherable after the sections are combined. Whenthe sections have equal length, the recipient deciphers the message using thenumber of sections or their length and the base.
It is necessary to use a base such that all of the numerical representations of the sections have the same length in that base. Few bases are capable offulfilling this requirement for a given set of sections. Trial and error can beused to find a base such that all the encoded sections have the same length.
In my research, I discovered an original equation which predicts the smallestinteger in a fractional base. This equation may be used to determine if allthe encrypted sections in that base are of equal length.
For instance, we have the message “ISEF Participants in Indiana” to encrypt using this method. We convert this message to the numerical repre-sentation 09190506001601182009030916011420190009140009140409011401 Evidently this would take a prohibitive amount of time to encode, so we splitit into sections. We attempt the base 729 : 729 is 272, thus every digit in base 729 can be represented by a combination of 2 characters.
My original equation for the smallest integer with a given length in an improper fractional base can be used to much advantage. We use it to findthat every base 729 integer with 6 digits is at least 12,885,896,162,367 in base 10, and every base 729 integer with 7 digits is at least 4,696,909,151,183,136 in base 10. Thus if we split the message into 16-digit sections, each of thesections will be equally long when encrypted: We may add some spaces - pairs of zeros - to the end of the message, tomake all the unencrypted sections the same length, so it is easier to decrypt.
Leading zeros on the sections are dropped, so if the recipient knows all thesections are the same length, the missing zeros can be replaced.
Now we convert each of these numbers to base 729 : We then convert each of these digits to their 2-digit base-27 representation: 05, 07; 11, 08; 18, 26; 26, 02; 00, 16; 07, 11 11, 15; 03, 09; 02, 21; 22, 07; 02, 26; 23, 11 10, 24; 17, 20; 08, 19; 02, 14; 21, 13; 23, 24 05, 05; 00, 20; 10, 05; 11, 23; 07, 05; 00, 07 We finally convert each of these digits into the corresponding character andjoin the message. As before, ‘00’ is equal to a space, ‘01’, ‘a’, etc.: ‘egkhrzzb pgkkocibuvgbzwkjxqthsbnumwxee tjekwge g’ To decrypt the message, the recipient must know that 729 was used as the base and that the number was split into 4 sections to encrypt. Even ifsomeone knew the base was 729 , the message is 48 characters long. There are 8 lengths possible to split up the message, but only one of these will givethe correct result.
Breaking a message encrypted in this manner requires testing possible basesand section lengths until the correct combination is found. The denominatormust divide the first digit of the fractionally based integer evenly, thus if thefirst digit is known, the possible denominators can be limited to divisors ofthat digit. The numerator of the fractional base must be at least as large as the number of unique digits in the numerical representation of the encryptedmessage, thus if the numerical representation of the encrypted message inthe improper fractional base is known, the numerator of the fractional baseis limited to a few values.
Using a base with a numerator greater than the number of characters available and using combinations of characters to represent the digits canmake it much more difficult to crack the message. As groups of charactersrepresent one digit, the first digit is much more difficult to find, thus findingthe correct denominator is also more difficult. For the same reason, it isharder to find the number of unique digits, thus making it also more difficultto narrow the possible numerators.
I created an original formula which predicted the smallest integer with a givenlength in a fractional base from the base and length, in accordance with thehypothesis. The formula was tested for 100 sets of data and is conjecturedto work for all fractional bases.
From this formula I created another original equation yielding the num- ber of integers with a specific length in an improper fractional base.I alsodiscovered a new method of encryption using fractional bases, in which myequation is useful.
Since fractional bases were discovered in 1936, little had been investigated beyond the method of integer conversion. The uses of fractional bases areopen to exploration and will receive continued attention in coming years.
I have grateful appreciation of my instructors at the Art of Problem Solv-ing Online School (artofproblemsolving.com) who have taught me to lovethe beauty of mathematics. To Richard Rusczyk, Mathew Crawford, DavidPatrick, and Naoki Sato, thank you.
I would also like to thank Dr. Katherine Zehender, Nicholas Zehender, Paul Wrayno, Dr. Tim Clark, Rocke Verser, and Adeel Khan for their helpfulcomments.
My parents, John and Ann Dorminy, deserve my thanks for allowing me the opportunity to explore math on my own timetable and for their constantsupport.
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