3x3 Magic Square of 7 Squares
Study 1

A Characterization of
3-Square Arithmetic Progressions
with a Fixed Starting Value
or
How to Find a 3x3 Magic Square of Squares
Having Monstrously Large Numbers

For the experts involved With problems, unsolved, A puzzle that stymies the best devotee: Magic square of squared numbers That further encumbers All entries, distinct, with its size, three by three. If computers should dare To seek such a square, A warning to program designers in charge: Martin Gardner insists That "If it exists, Its numbers are sure to be monstrously large".
Introduction Results of the Study Part 1 - Generating All Arithmetic Progressions The Arithmetic Progression Formula Forward Recursion Theorem Reverse Recursion Formulas 5/7 Lemma Reverse Recursion Reduction Lemma Finite Generator Theorem Generator Nonredundancy Theorem Relative Placement Theorem Part 2 - Enumerating Potential Magic Squares Magic Square Requirements Magic Square Enumeration Procedure Changing Geometric Progression Lemma Corresponding Combination Rejection Lemma Enumeration Termination Theorem Part 3 - Finding Generators Prime Factor Reduction Formula for the Number of Generators Generators Come In Pairs The Pre-Generator Method Composition of Forms Finding Pre-Generators For Primes Part 4 - Future Research Extending the Results Magic Square Search Ideas Impossibility Proof Ideas Introduction A necessary condition for a 3x3 magic square of distinct squares is a solution to any of its 7-square subsets. This paper studies the following 7-square subset where a,b,c > 0 and thus, c-(a+b) is the smallest entry. ------- c-(a+b) ------- c+(b-a) c c-(b-a) c-b c+(a+b) c-a This configuration contains three arithmetic progressions having the same starting value: c-(a+b), c-b, c-(b-a), with step value a; c-(a+b), c-a, c+(b-a), with step value b; and c-(a+b), c, c+(a+b), with step value a+b. Since these values must be squares, this motivates the study of 3-square arithmetic progressions having a fixed starting value. If we can find a set of three progressions that also have the step value relationship (a, b, a+b), we will have found a magic square containing at least 7 squared entries. Suppose we have a 3-square arithmetic progression given by L2, Mn2, Hn2 where 0 < L < Mn < Hn, and suppose that the smallest value, L = 1. Here is a list of the progressions for n = 1 ... 5. n L2 Mn2 Hn2 STEP --- --- ----- ----- ------- 1 12 52 72 24 2 12 292 412 840 3 12 1692 2392 28560 4 12 9852 13932 970224 5 12 57412 81192 32959080 Every 3-square arithmetic progression starting from 1 can be produced using a simple recursion. Note how fast the step values increase in size. If there were a magic square having 1 as the smallest value, but with the largest value having 100 digits, that solution could be found in about 66 iterations of the recursion. But because any choice of the largest step value is more than twice the size of any step value below it, there can't be a selection of three step values with the relationship (a, b, a+b). Therefore L = 1 cannot be the smallest entry in this 7-square configuration. This also means that 1 cannot be the smallest entry in any 3x3 magic square of distinct squares. Other starting values can be proved impossible using a termination condition, derived in this paper. It has been found that the termination condition occurs surprisingly early in a generated sequence of candidate step values. So the impossibility of each starting value takes very little time to prove. Results of the Study This paper contains a proof that all 3-square arithmetic progressions with a fixed starting value can be produced by simple recursions of an easily obtainable finite set of generators. Each recursively produced sequence consists of values in a near-geometric progression increasing by about 34 times for each iteration, so the numbers get very large very fast. This makes it possible to do a quick but complete search for magic squares having monstrously large numbers. But better than that, this paper contains a proof that if an enumeration is done for a particular starting value, then once a certain condition occurs, it is no longer possible to achieve a solution, and so the enumeration can be terminated as impossible to satisfy. Enumerations have been done and it has been found that there is no solution to the above 7-square configuration having a starting value less than 1014. It has been known for a long time that a 3x3 magic square of squares must contain very large numbers, but the current result is new in that all the numbers must be large. All nine entries of a 3x3 magic square of distinct squares must be at least the squares of 8-digit numbers. Part 1 Generating All Arithmetic Progressions This part of the paper develops and proves a technique for generating all possible 3-square arithmetic progressions having a fixed starting value. The technique uses a recursive formula which takes a "generator" progression and produces an infinite series of progressions with increasing values. The Arithmetic Progression Formula The 3-square arithmetic progression under study is L2, Mn2, Hn2, where L is fixed. The step value of the progression = Mn2 - L2 = Hn2 - Mn2. This is expressed in this paper as (1) L2 = 2Mn2 - Hn2; 0 < L < Mn < Hn. We want to represent a given value of L2 as this binary quadratic form in all possible ways. We also want to generate those representations, as needed, in Hn order: H1 < H2 < H3, and so on. The recursion described below will accomplish this, however, it must be noted that putting a representation using (Mn,Hn) into the recursion does not in general produce (Mn+1,Hn+1). For example, if L is prime, then putting (Mn,Hn) into the recursion will produce (Mn+3,Hn+3). Thus, in this case we need three different recursive sequences in order to produce all the representations. In this paper, the number of different recursive sequences needed to produce all Hn values for a given L is given by g. That is, putting (Mn,Hn) into the recursion produces (Mn+g,Hn+g). Forward Recursion Theorem Given a representation of (1) using (Mn,Hn) for a given L, then (Mn+g,Hn+g) given by (2a) Mn+g = 3Mn + 2Hn, (2b) Hn+g = 3Hn + 4Mn is also a representation. Proof Starting with 2Mn+g2 - Hn+g2, substitute (2a), (2b), 2(3Mn + 2Hn)2 - (3Hn + 4Mn)2, expand the terms, 18Mn2 + 8Hn2 + 24HnMn - (9Hn2 + 16Mn2 + 24HnMn), and cancel, producing 2Mn2 - Hn2, which from (1) is equal to L2. Thus 2Mn+g2 - Hn+g2 = L2. Remark Note that the recursion has only positive coefficients, therefore starting with a representation using positive numbers, the recursion produces a sequence of representations using increasingly larger positive numbers. Example The number of different recursion sequences that can be produced for a given L is given by g. Suppose that L = 7. It will be found that g = 3. The smallest representation uses (M1,H1) = ( 13, 17); 72 = 2(132) - 172. Putting these values into the recursion produces (M4,H4) = ( 73,103); 72 = 2(732) - 1032; (M7,H7) = (425,601); 72 = 2(4252) - 6012. Here is the second smallest representation for L = 7. It produces a different sequence. (M2,H2) = ( 17, 23); 72 = 2(172) - 232. Putting these values into the recursion produces (M5,H5) = ( 97,137); 72 = 2(972) - 1372; (M8,H8) = (565,799); 72 = 2(5652) - 7992. Here is the third smallest representation where Hg = 7L. (M3,H3) = ( 35, 49); 72 = 2(352) - 492. Putting these values into the recursion produces (M6,H6) = ( 203, 287); 72 = 2(2032) - 2872; (M9,H9) = (1183,1673); 72 = 2(11832) - 16732. Remark Note that H1 through H9 are in increasing order, even though they came from multiple sequences. It will be proved below that this will always be true for any L. Reverse Recursion Formulas Solving for the inverse recursion formulas from (2a), (2b), (3a) Mn = 3Mn+g - 2Hn+g, (3b) Hn = 3Hn+g - 4Mn+g. Remark Since the forward recursion takes positive values and produces a sequence of ever-increasing values, the reverse recursion must produce a sequence of ever-decreasing values. 5/7 Lemma Given a representation of (1) using (Mn,Hn) for a given L, if Hn > 7L, then Mn < (5/7)Hn. Proof If Hn > 7L then L < Hn/7 and combining with (1) 2Mn2 = Hn2 + L2 < Hn2 + Hn2/49 or Mn2 < (25/49)Hn2 or Mn < (5/7)Hn. Reverse Recursion Reduction Lemma Given a representation of (1) using (Mn+g,Hn+g) for a given L, then applying the reverse recursion (3a),(3b) to produce (Mn,Hn), if Hn+g > 7L, then Hn > L. Proof If Hn+g > 7L, then from the 5/7 Lemma Mn+g < (5/7)Hn+g. Combining with (3a), Hn = 3Hn+g - 4Mn+g > 3Hn+g - 4(5/7)Hn+g or Hn > Hn+g/7. And since Hn+g > 7L, Hn > L. Finite Generator Theorem A generator is a representation of (1) that is not produced by any other representation using the recursion (2a),(2b). For a given value of L, there is a finite number, g, of generators using (M1,H1) ... (Mg,Hg) and L < H1 < ... < Hg < 7L. Proof Starting from any representation where Hn > 7L and applying the reverse recursion (3a),(3b) repeatedly, we will eventually encounter a terminal representation with Ht < 7L, for some t. When this happens, the previous representation that produced it must have had Ht+g > 7L, thus this terminal representation must have Ht > L by the Reverse Recursion Reduction Lemma. Therefore, this terminal representation has L < Ht < 7L, and is the generator of the sequence of representations that were encountered using the above reverse recursion procedure. When the forward recursion (2a),(2b) is applied, all representations in that sequence are reachable. Since starting from any representation and applying the reverse recursion always eventually produces a representation with L < Ht < 7L, then all representations with numbers greater than that range are reachable by starting with a representation in that range and applying the forward recursion. Since that range is finite for a given value of L, the number of generators is finite for a given L. Remark Finding all generators is just a matter of testing values for Hn, n = 1 ... g-1, in the range L ... 7L that satisfy (1); then adding Hg = 7L, which is always a generator. Part 3 of this paper shows that there exist faster ways. Generator Nonredundancy Theorem All representations using (Mn,Hn) with L < Hn < 7L are generators. Remark When searching for generators in the finite range, there is no chance that one of them could be produced by another in the same range. Therefore, in a computer procedure, there is no need to check for duplicate values. Proof Given a representation using (Hn,Mn) with L < Hn < 7L, then from (1), 2Mn2 = Hn2 + L2 > 2L2 thus Mn > L. From the recursion (2a), Hn+g = 3Hn + 4Mn and since 3Hn + 4Mn > 3L + 4L = 7L we have Hn+g > 7L. Therefore, one application of the recursion on a generator always produces a representation past the range of the generators. So there is no chance of finding a non-generator in the generator range. Relative Placement Theorem Any two different recursive sequences produce mutually exclusive sets of representations and their relative placements within the sets of sequences never change. This is because Hj < Hk implies Hj+g < Hk+g for any j and k. Remark This is important for the orderly enumeration of representations. The example above with L = 7 and g = 3 shows that the Hn values are produced in increasing order by applying the recursion once for each generator sequence, then repeating the recursion once each for the produced values, and so on. Proof Suppose (Mj,Hj) and (Mk,Hk) are used in two different representations produced from possibly different generated sequences with Hj < Hk. We have 2Mj2 - Hj2 = L2, 2Mk2 - Hk2 = L2, thus 2Mj2 = Hj2 + L2 and since Hj < Hk 2Mj2 < Hk2 + L2 or 2Mj2 < 2Mk2 or Mj < Mk. Therefore, Hj < Hk implies Mj < Mk. Suppose that g is the number of generators and the forward recursion (2a),(2b) produces Hj+g from Hj and Hk+g from Hk in the sets of sequences. Then Hj+g = 3Hj + 4Mj and since Hj < Hk and Mj < Mk Hj+g < 3Hk + 4Mk or Hj+g < Hk+g. Therefore, Hj < Hk implies Hj+g < Hk+g for any j and k, and their relative placement never changes. This also means that their values can never be the same, thus multiple recursive sequences produce mutually exclusive values. Part 2 Enumerating Potential Magic Squares This part of the paper describes a procedure for enumerating a sequence of 3-square arithmetic progressions in order to find a magic square. When certain conditions occur in the sequence, the procedure terminates because it is no longer possible to satisfy the magic square requirements. These termination conditions are proved below. Magic Square Requirements See the Introduction to this paper for a description of the 7-square subset of the 3x3 magic square of distinct squares that we are studying. For a solution to this configuration, we need to find three 3-square arithmetic progressions having the same starting value and their step values must have the relationship (a, b, a+b). In other words, the sum of the step values of the first two arithmetic progressions equals the step value of the third. We can also express this with twice the step values. Twice the step value of arithmetic progression L2, Mn2, Hn2 is (Hn2 - L2). So we need to find three representations of L2 that use (Mi,Hi), (Mj,Hj), (Mk,Hk) and that satisfy the equation (Hi2 - L2) + (Hj2 - L2) = (Hk2 - L2). Note that Hk must be the largest term. Adding 2L2 to both sides gives (4) Hi2 + Hj2 = Hk2 + L2; Hi < Hj < Hk. Magic Square Enumeration Procedure After finding the generators (M1,H1) ... (Mg,Hg), arranged in increasing Hn value, then the rest of the representations can be produced in order, one at a time, and successive values of Hn2 put into a list. The increasing order is guaranteed by the Relative Placement Theorem. Each a time a new Hn2 is produced, it can be checked with combinations of Hn2 values that come before it in that list to try and satisfy (4). Checking is quicker than you might think. This is because, most of the time, Hn-22 + Hn-12 < Hn2 + L2. This means that the biggest sum that can be made using previous Hn values is too small. So, after just a single check, all possible combinations using the latest Hn can be rejected. It will also be seen below that if the sum is too small by 2L2, that is, Hn-22 + Hn-12 < Hn2 - L2, then Hn+g, Hn+2g, etc. can also be rejected. When this simple case does not occur, checking combinations using Hn can be done in at worst linear time in the length of the list. Here is a general procedure. Set k := n; i := 1; j := k-1 Do Compare Hi2 + Hj2 to Hk2 + L2; If too small, increment i; Else If too large, decrement j; Else just right; (you have found a magic square of 7 squares) While i < j (which will be at most k-2 times) Example Let's use the L = 7 numbers above. n 1 2 3 4 5 6 7 8 9 Hn 17 23 49 103 137 287 601 799 1673 i=1,j=2,k=3 H32 + L2 = 492 + 72 = 2450. H12 + H22 = 172 + 232 = 818, too small by more than 2x72. i=2,j=3,k=4 H42 + L2 = 1032 + 72 = 10658. H22 + H32 = 232 + 492 = 2930, too small by more than 2x72, so it's not necessary to check i=1 because the sum will be even smaller. i=3,j=4,k=5 H52 + L2 = 1372 + 72 = 18818. H22 + H32 = 492 + 1032 = 13010, too small by more than 2x72, so it's not necessary to check i=1 or i=2 because the sum will be even smaller. As the following proofs will show, we need not go any further. Since we have 3 consecutive rejections (g = 3) and conditions (5) and (6) below are met for each rejection, all the rest of the H values will be rejected in the same way. We have proved that 72 cannot be the smallest entry in any 3x3 magic square of distinct squares. Changing Geometric Progression Lemma If (Mj,Hj) and (Mk,Hk) with j < k are used in (1), then Hj+g/Hj > Hk+g/Hk and Mj+g/Mj < Mk+g/Mk. Proof If (Mj,Hj) and (Mk,Hk) with j < k are used in (1), then Hj < Hk thus L2/Hj2 > L2/Hk2. Dividing (1) by Hn2 gives L2/Hn2 = 2Mn2/Hn2 - 1. Substituting j and k for n and combining the above result, 2Mj2/Hj2 - 1 > 2Mk2/Hk2 - 1 or Mj/Hj > Mk/Hk. Using the forward recursion formula (2b) and dividing by Hn, Hn+g/Hn = 3 + 4Mn/Hn. Substituting j and k for n and combining the above result, Hj+g/Hj = 3 + 4Mj/Hj > 3 + 4Mk/Hk = Hk+g/Hk, Therefore Hj+g/Hj > Hk+g/Hk. Repeating the derived condition Mj/Hj > Mk/Hk we also have Hj/Mj < Hk/Mk. Using the forward recursion formula (2a) and dividing by Mn, Mn+g/Mn = 3 + 2Hn/Mn. Substituting j and k for n and combining the above result, Mj+g/Mj = 3 + 2Hj/Mj < 3 + 2Hk/Mk = Mk+g/Mk. Therefore Mj+g/Mj < Mk+g/Mk. Corresponding Combination Rejection Lemma Given the values Hi, Hj, Hk from a sequence of representations of (1) for a given L, with i < j < k, (Case 1) if Hi2 + Hj2 < Hk2 - L2 then Hi+g2 + Hj+g2 < Hk+g2 - L2; (Case 2) if Hi2 + Hj2 > Hk2 + L2 then Hi+g2 + Hj+g2 > Hk+g2 + L2. Remark This means that once one of the above two cases becomes true, it will also be true for the corresponding combination of values with index offsets of multiples of g. That is, if it's true for the combination (Hi,Hj,Hk), then it will be true for (Hi+g,Hj+g,Hk+g), (Hi+2g,Hj+2g,Hk+2g), and so on. All of them will produce a sum outside of the interval Hk2 ± L2, and therefore can never satisfy (4). Therefore, it will no longer be necessary to check those combinations in the rest of the enumeration for a given value of L. Proof (Case 1) If Hi2 + Hj2 < Hk2 - L2, then adding 2L2 to both sides and using (1) we get 2Mi2 + 2Mj2 < 2Mk2. Multiplying by (Mk+g/Mk)2, 2Mi2(Mk+g/Mk)2 + 2Mj2(Mk+g/Mk)2 < 2Mk+g2. From the Changing Geometric Progression Lemma with j < k, Mj+g/Mj < Mk+g/Mk. Squaring and rearranging, we get Mj+g2 < Mj2(Mk+g/Mk)2. Similarly, with i < k, Mi+g2 < Mi2(Mk+g/Mk)2. Combining the above results, 2Mi+g2 + 2Mj+g2 < 2Mk+g2 and then subtracting 2L2 from both sides and using (1), Hi+g2 + Hj+g2 < Hk+g2 - L2. (Case 2) If Hi2 + Hj2 > Hk2 + L2, then multiplying by (Hk+g/Hk)2, we get Hi2(Hk+g/Hk)2 + Hj2(Hk+g/Hk)2 > Hk+g2 + L2(Hk+g/Hk)2. From the Changing Geometric Progression Lemma with j < k, Hj+g/Hj > Hk+g/Hk. Squaring and rearranging, we get Hj+g2 > Hj2(Hk+g/Hk)2. Similarly, with i < k, Hi+g2 > Hi2(Hk+g/Hk)2. Combining the above results, Hi+g2 + Hj+g2 > Hk+g2 + L2(Hk+g/Hk)2, and since Hk+g/Hk > 1, Hi+g2 + Hj+g2 > Hk+g2 + L2. Enumeration Termination Theorem If for g consecutive values of Hk, (r+1 < k < r+g), for some r, (5) H12 + Hk-12 < Hk2 - L2 and for each combination of Hi and Hj, i < j < k, either (6) Hi2 + Hj2 < Hk2 - L2 or (7) Hi2 + Hj2 > Hk2 + L2, then it is impossible for any Hi, Hj, Hk combination where i < j < k and k > r+1, to satisfy (4) and the magic square enumeration for the given value of L can be terminated. Proof If (6) or (7) applies, then we know from the Corresponding Combination Rejection Lemma that all future corresponding combinations will not satisfy (4). But this doesn't cover all possible future combinations. New combinations will be created because there are more representations in the list. For example, suppose (7) is met using i=2,j=k-1; the sum is too big. Then we also know that (7) is met using i=2+g,j=k-1+g. But then, new combinations where i=1...g+1 and j=k-1+g could have a smaller sum. These would need to be tested -- unless you knew that (5) was met. Then all new combinations would have a sum which was too small. Therefore, after g consecutive rejections using the above criteria, all combinations, old and new, can be rejected. Example Just for completeness, let's try the enumeration for L = 1, to prove that the comments in the Introduction are true. Only the first three representations are needed because g = 1. n 1 2 3 Hn 7 41 239 i=1,j=2,k=3 H32 - L2 = 2392 - 12 = 57,120. H12 + H22 = 72 + 412 = 1730. Conditions (5) and (6) are met and since g = 1, we are done. Thus, 1 cannot be the lowest entry in any 3x3 magic square of distinct squares. Example L = 41 is the smallest value of L where condition (7) appears. Since L is prime, g = 3. We only need the first five representations. n 1 2 3 4 5 Hn 113 119 287 679 713 i=1,j=2,k=3 H32 - L2 = 2872 - 412 = 80,688. H12 + H22 = 1132 + 1192 = 26,930. Conditions (5) and (6) are met. i=2,j=3,k=4 H42 - L2 = 6792 - 412 = 459,360. H22 + H32 = 1192 + 2872 = 96,530. Condition (6) is met for the largest values of i and j, therefore condition (6) is met for all combinations, condition (5) being one of them. i=3,j=4,k=5 H52 + L2 = 7132 + 412 = 510,050. H32 + H42 = 2872 + 6792 = 543,410. Condition (7) is met, so we need to test k=5 further. i=2,j=4,k=5 H52 - L2 = 7132 - 412 = 506,688 H22 + H42 = 1192 + 6792 = 475,202. Condition (6) is met. All the rest of the combinations will have lower sums than this one, so they all satisfy condition (6). Condition (5) is also satisfied. We have 3 consecutive rejections of all combinations, so 412 cannot be the lowest entry in any 3x3 magic square of distinct squares. Example L = 71 is the smallest value of L where a value of Hk cannot be rejected and more representations have to be generated. Since L is prime, g = 3. n 1 2 3 4 5 6 7 Hn 97 391 497 631 2297 2911 3689 i=1,j=2,k=3 H32 - L2 = 4972 - 712 = 241,968. H12 + H22 = 972 + 3912 = 162,290. Conditions (5) and (6) are met. i=2,j=3,k=4 H42 - L2 = 6312 - 712 = 393,120. H42 + L2 = 6312 + 712 = 403,202. H22 + H32 = 3912 + 4972 = 399,890. Neither condition (6) or (7) is met, so we can't reject this combination. Since the sum is too small for (4) to be satisfied, there's no need to try i=1,j=3 or i=1,j=2, which would make an even smaller sum. i=3,j=4,k=5 H52 - L2 = 22972 - 712 = 5,271,168. H32 + H42 = 4972 + 6312 = 645,170. Condition (6) is met and thus is met by all other combinations including condition (5). k=6 Since all combinations of k=3 were rejected with (5) and (6), all future combinations of k=6, k=9, etc. are also rejected and need not be tested. i=5,j=6,k=7 H72 + L2 = 36892 + 712 = 13,613,762. H52 + H62 = 22972 + 29112 = 13,750,130. Condition (7) is met, so we need to test k=7 further. i=4,j=6,k=7 H72 - L2 = 36892 - 712 = 13,603,680. H42 + H62 = 6312 + 29112 = 8,872,082. Condition (6) is met. All the rest of the combinations will have lower sums than this one, so they all satisfy condition (6). Condition (5) is also satisfied. We now have 3 consecutive rejections of all combinations, so 712 cannot be the lowest entry in any 3x3 magic square of distinct squares. Example L = 49 is not a prime and g = 5. n 1 2 3 4 5 6 7 Hn 71 119 161 257 343 457 721 The first 5 tests all meet conditions (5) and (6), so we have 5 consecutive rejections of all combinations. Example L = 119 is not a prime and g = 9. It will be found that k = 3,4,6,7,8 can't be rejected, but k = 9 through 17 are rejected, making 9 consecutive rejections. Part 3 Finding Generators This part of the paper describes efficient methods for finding generators. For large values of L, finding all the generators can be more time-consuming than doing the calculations to search for the magic square requirements and to reject combinations. Prime Factor Reduction In a 3-square arithmetic progression, if either the lowest or highest values have a prime factor of the form 8k+3 or 8k+5, then all three terms will have that factor. This also means that all 7 of the entries in a magic square will have that factor. So we can divide out the factor producing a smaller magic square. Therefore it is not necessary to test values of L and H having those primes as factors. This leaves values of L and H having prime factors of only 8k+1 and 8k+7. The first of these numbers are 1, 7, 17, 23, 31, 41, 47, 49, 71, 73, 79, 89, 97, 103, 113, 119. This reduces the number of L values that need to be tested and also reduces the number of H values that need to be searched to find the generators. They both come from the same list. Proof Given L2 = 2M2 - H2, we factor this in the unique factorization domain Z[√2] as L2 = (M√2 + H)(M√2 - H). If a prime of the form 8k+3 or 8k+5, which is also a prime in the domain Z[√2], is a factor of L, then it must also be a factor of (M√2 + H) or (M√2 - H). If it is a factor of one of them, it is also a factor of the conjugate, so it is a factor of both. If it is a factor of both, then it is also a factor of their difference, which is 2H. Since the factor is odd, it is a factor of L. If it is a factor of H and L and odd, then it is also a factor of M. Thus, it is a factor of all three terms and can be factored out to produce a smaller solution. Therefore, values of L having prime factors of 8k+3 or 8k+5 need not be tested. Formula for the Number of Generators If the prime factorization of L = paqb...rc, then the number of generators g = (2a + 1)(2b + 1) ... (2c + 1). Examples For L = 1, g = 1. Given that p, q, r are primes: If L = p, then g = 3. If L = p2, then g = 5. If L = p3, then g = 7. If L = pq, then g = 9. If L = p2q, then g = 15. If L = pqr, then g = 27. Proof Generators Come In Pairs If (Mn,Hn) is a generator for a given L, then so is (Mp,Hp) given by Mp = 17Mn - 12Hn, Hp = 24Mn - 17Hn. Proof We have to prove that (Mp,Hp) gives a representation of L2 using (1). Starting with 2Mp2 - Hp2 and substituting the relation above 2(17Mn - 12Hn)2 - (24Mn - 17Hn)2 expand the terms and cancel, producing 2Mn2 - Hn2 which is equal to L2. We also have to prove that the formula produces generators from other generators. If (Mn,Hn) is a generator, then their values range from M = L, H = L to M = 5L, H = 7L. Thus Hp ranges between 24L - 17L and 24x5L - 17x7L or 7L and L, the same range as Hn when it is a generator. Also, the case where Hp = Hn doesn't exist since if Hn = Hp = 24Mn - 17Hn. then (3/4)Hn = Mn and putting this into (1) gives 2(9/16)Hn2 - Hn2 = Ln2 or Hn2 = 8L2 or Hn = L√8 which can never be an integer. The Pre-Generator Method In the formula for the number of generators, g is always an odd number. But Hg = 7L is always a generator. So that leaves an even number of other generators between L and 7L. As the above proof shows, these other generators come in related pairs. Suppose that we have a complete set of generators for a given L, Applying the reverse recursion to these values produces all values less than L so that Hn-g < Mn-g < L. All Mn-g values will be positive, but exactly half of the Hn-g values will be negative. If Hn > L√8, then Hn-g > 0. If Hn < L√8, then Hn-g < 0. If (Mn-g,-Hn-g) satisfies (1), then so does (Mn-g,Hn-g). This means that the negative values of Hn-g and positive values of Hn-g come in pairs of equal absolute value. Therefore, we can do a faster search for generators by looking for values of Hn-g in the pre-generator range 0 < Hn-g < L, which is 1/6 the range of L < Hn < 7L. For each of these pre-generators, we list both positive and negative values of Hn-g. We then put the pre-generators into the forward recursion to get the generators. Then we add Hg = 7L to the list. The Prime Factor Reduction also applies to these Hn-g values. It is sufficient to list values of Hn-g that have prime factors of only 8k+1 and 8k+7. Example For L = 7, g = 3. In the range 1 ... 7, we find 1 pre-generator. Listing both the negative and positive pre-generators. n 1 2 Hn-g -1 1 Mn-g 5 5 Putting these through the forward recursion and adding 7L produces n 1 2 3 Hn 17 23 49 Example For L = 119 = 7x17, g = 9. In the range 1 ... 119, we find 4 pre-generators. Listing both the negative and positive pre-generators. n 1 2 3 4 5 6 7 8 Hn-g -79 -49 -41 -17 17 41 49 79 Mn-g 101 91 89 85 85 89 91 101 Putting these through the forward recursion and adding 7L produces n 1 2 3 4 5 6 7 8 9 Hn 167 217 233 289 391 479 511 641 833 Composition of Forms Suppose we have already tested Lu and Lv and have saved the list of pre-generators for each. We then come to Lw = LuLv. We can directly compute all the pre-generators for Lw using the lists of pre-generators for Lu and Lv. To do this we use the following composition and scaling formulas. (8a) Mw = (2MuMv + HuHv) - (MuHv + HuMv) (8b) Hw = |(2MuMv + HuHv) - 2(MuHv + HuMv)| (9a) Mw = (2MuMv - HuHv) - |MuHv - HuMv| (9b) Hw = (2MuMv - HuHv) - 2|MuHv - HuMv| (10a) Mw = MuLv (10b) Hw = HuLv (11a) Mw = MvLu (11b) Hw = HvLu To use the above formulas for the case where Lu and Lv have no prime factors in common, follow this procedure. For each (Mu,Hu), For each (Mv,Hv), Use (8a),(8b); Use (9a),(9b). For each Mu,Hu), Use (10a),(10b). For each (Mv,Hv), Use (11a),(11b). If Lw is a power of a prime, pa, then its pre-generators can be computed from the pre-generators for Lu = p and Lv = pa-1. The procedure is different, depending on whether the power is even or odd. It is also required to keep track of the one primitive pre-generator in each list of pre-generators for prime powers. For a prime, there is exactly one pre-generator and it is primitive. For a prime power, follow this procedure. To compute the scaled pre-generators of Lw: For each (Mv,Hv), Use (11a),(11b). To compute the primitive pre-generator of Lw: If the power, a, is even Use (8a),(8b) on both primitive pre-generators. Else Use (9a),(9b) on both primitive pre-generators. Example Lw = 72. For Lu = 7, (Mu,Hu) = {(5,1)} and is primitive. For Lv = 7, (Mv,Hv) = {(5,1)} and is primitive. The power of the prime is even, so we use (8a),(8b) and (11a),(11b). Lu Mu Hu Mv Hv Mw Hw (8a), (8b) --> -- 5 1 5 1 41 31 (primitive) (11a),(11b) --> 7 - - 5 1 35 7 Example Lw = 73. For Lu = 7, (Mu,Hu) = {(5,1)} and is primitive. For Lv = 72, (Mv,Hv) = {(35,7), (41,31)}, the last one being primitive. The power of the prime is odd, so we use (9a),(9b) and (11a),(11b). Lu Mu Hu Mv Hv Mw Hw (9a), (9b) --> -- 5 1 41 31 265 151 (primitive) (11a),(11b) --> 7 - - 41 31 287 217 (11a),(11b) --> 7 - - 35 7 245 49 Example Lw = 391 = 17x23 For Lu = 23, (Mu,Hu) = {(17,7)}. For Lv = 17, (Mv,Hv) = {(13,7)}. Lu Lv Mu Hu Mv Hv Mw Hw (8a), (8b) --> -- -- 17 7 13 7 281 71 (9a), (9b) --> -- -- 17 7 13 7 365 337 (10a),(10b) --> -- 17 17 7 -- - 289 119 (11a),(11b) --> 23 -- - - 13 7 299 161 Example Lw = 19159 = 49x391 = 72x(17x23) For Lu = 49, (Mu,Hu) = {(41,31),(35,7)}. For Lv = 391, (Mv,Hv) = {(281,71),(289,119),(299,161),(365,337)}. Lu Lv Mu Hu Mv Hv Mw Hw (8a), (8b) --> -- --- 41 31 281 71 13621 1999 (8a), (8b) --> -- --- 41 31 289 119 13549 289 (8a), (8b) --> -- --- 41 31 299 161 13639 2231 (8a), (8b) --> -- --- 41 31 365 337 15245 9887 (8a), (8b) --> -- --- 35 7 281 71 15715 11263 (8a), (8b) --> -- --- 35 7 289 119 14875 8687 (8a), (8b) --> -- --- 35 7 299 161 14329 6601 (8a), (8b) --> -- --- 35 7 365 337 13559 791 (9a), (9b) --> -- --- 41 31 281 71 15041 9241 (9a), (9b) --> -- --- 41 31 289 119 15929 11849 (9a), (9b) --> -- --- 41 31 299 161 16859 14191 (9a), (9b) --> -- --- 41 31 365 337 16981 14479 (9a), (9b) --> -- --- 35 7 281 71 18655 18137 (9a), (9b) --> -- --- 35 7 289 119 17255 15113 (9a), (9b) --> -- --- 35 7 299 161 16261 12719 (9a), (9b) --> -- --- 35 7 365 337 13951 4711 (10a),(10b) --> -- 391 41 31 -- -- 16031 12121 (10a),(10b) --> -- 391 35 7 -- -- 13685 2737 (11a),(11b) --> 49 --- - - 281 71 13769 3479 (11a),(11b) --> 49 --- - - 289 119 14161 5831 (11a),(11b) --> 49 --- - - 299 161 14651 7889 (11a),(11b) --> 49 --- - - 365 337 17885 16513 Finding Pre-Generators For Primes The Pre-Generator Method reduces the problem from scanning for generators in the range L ... 7L to instead scanning in the 1/6 size range 0 ... L. The Composition of Forms reduces the problem to scanning for pre-generators only for primes, while pre-generators for composites are directly computed. This section shows how to further reduce the work in finding the pre-generator for a prime by scanning in the much smaller range 0 ... √L. Also, since there is only one pre-generator for a prime, once you find it, you can stop scanning. Instead of looking for a representation for L2, look for the representation, L = 2m2 - h2, then use a version of the composition formulas to compute the pre-generator for L2. (12a) M = 2m2 + h2 - 2mh (12b) H = |2m2 + h2 - 4mh| Examples L m h M H L m h M H 7 2 1 5 1 89 7 3 65 23 17 3 1 13 7 97 7 1 85 71 23 4 3 17 7 103 8 5 73 7 31 4 1 25 17 113 9 7 85 41 41 5 3 29 1 127 8 1 113 97 47 6 5 37 23 137 9 5 97 7 71 6 1 61 49 151 10 7 109 31 73 7 5 53 17 191 10 3 149 89 79 8 7 65 47 199 10 1 181 161 Remark Here are some observations that lead to some extra time savings. All the h values are odd. An m value is odd when L = 8k+1. An m value is even when L = 8k+7. Will this always be true? Proof We have h2 = 2m2 - L. 2m2 is even and L is odd, therefore h must be odd. The square of an odd number has the form 8k+1. The square of an even number has the form 8k or 8k+4. We have 2m2 = L + h2. Since h is odd, h2 has the form 8k+1. If L has the form 8k+1, then 2m2 must have the form 8k+2, and m must be odd. If L has the form 8k+7, then 2m2 must have the form 8k, and m must be even. Part 4 Future Research This part of the paper contains ideas for extending the results to larger numbers, determining places where a magic square might be hiding, and for proving the impossiblity of the full 9-square problem. Extending the Results The enumeration described in the Introduction was performed using the Pre-Generator Method, but without the Composition of Forms or √L Prime scanning. Thus, looking for pre-generators consumed most of the time. So the enumeration was aborted after examining values of L up to 7 digits. Using the new methods should speed up the operation enough so that much larger values for L can be tested. A new implementation would require a technique to build values of L from a list of 8k+1/8k+7 primes, similar to the technique that Christian Boyer used in his search. I look forward to an independent verification of my results and a possible extension of it, possibly even finding a magic square. Magic Square Search Ideas Here are some ideas for trying various values of L to try and find a magic square without enumerating everything. The best value of L to use for finding a magic square would be one that had a lot of different small prime factors. This is because the density of H values would be greatest, increasing the chance for (4) to be met. Using higher powers of the same prime does not increase the density as much as different primes. This can be seen from the formula for the number of generators. A procedure to search for a magic square is to try values of L in the following sequence, 7, 7x17, 7x17x23, 7x17x23x31, etc. including the next prime for each search. The idea is that if there is a solution using a subset of these primes, then you would find a scaled version of the solution. So there is no need to try every subset when you can do them all at once. Make sure you have your giant integer arithmetic package ready. Impossibility Proof Ideas In order to prove that the full magic square of 9 squares is impossible, it is sufficient to prove restrictions on its 7-square subsets. If the restrictions are mutually exclusive, then they can't coexist in a 9-square solution at the same time. The results of this study suggest a restriction conjecture because the termination condition occurs so early. If it can be proved that the termination will occur before H3g, for example, then there is an upper limit to the ratio of the highest entry to the lowest entry. If there exists another 7-square configuration, which also includes the highest and lowest entries from the 9-square problem, and it shows that there is a lower limit to this same ratio that is greater than the upper limit, then the 9-square problem would be proved impossible.