Searching for RH Counterexamples — Search Strategies

We’re glibly searching for counterexamples to the Riemann Hypothesis, to trick you into learning about software engineering principles. In the first two articles we configured a testing framework and showed how to hide implementation choices behind an interface. Next, we’ll improve the algorithm’s core routine. As before, I’ll link to specific git commits in the final code repository to show how the project evolves.

Superabundant numbers

A superabundant number n is one which has “maximal relative divisor sums” in the following sense: for all m < n,

\displaystyle \frac{\sigma(m)}{m} < \frac{\sigma(n)}{n}

where \sigma(n) is the sum of the divisors of n.

Erdős and Alaoglu proved in 1944 (“On highly composite and similar numbers“) that superabundant numbers have a specific prime decomposition, in which all initial primes occur with non-increasing exponents

\displaystyle n = \prod_{i=1}^k (p_i)^{a_i},

where p_i is the i-th prime, and a_1 \geq a_2 \geq \dots \geq a_k \geq 1. With two exceptions (n=4, 36), a_k  = 1.

Here’s a rough justification for why superabundant numbers should have a decomposition like this. If you want a number with many divisors (compared to the size of the number), you want to pack as many combinations of small primes into the decomposition of your number as possible. Using all 2’s leads to not enough combinations—only m+1 divisors for 2^m—but using 2′ and 3’s you get (r+1)(s+1) for 2^r3^s. Using more 3’s trades off a larger number n for the benefit of a larger \sigma(n) (up to r=s). The balance between getting more distinct factor combinations and a larger n favors packing the primes in there.

Though numbers of this form are not necessarily superabundant, this gives us an enumeration strategy better than trying all numbers. Enumerate over tuples corresponding to the exponents of the prime decomposition (non-increasing lists of integers), and save those primes to make it easier to compute the divisor sum.

Non-increasing lists of integers can be enumerated in the order of their sum, and for each sum N, the set of non-increasing lists of integers summing to N is called the partitions of N. There is a simple algorithm to compute them, implemented in this commit. Note this does not enumerate them in order of the magnitude of the number \prod_{i=1}^k (p_i)^{a_i}.

The implementation for the prime-factorization-based divisor sum computation is in this commit. In addition, to show some alternative methods of testing, we used the hypothesis library to autogenerate tests. It chooses a random (limited size) prime factorization, and compares the prime-factorization-based algorithm to the naive algorithm. There’s a bit of setup code involved, but as a result we get dozens of tests and more confidence it’s right.

Search Strategies

We now have two search strategies over the space of natural numbers, though one is obviously better. We may come up with a third, so it makes sense to separate the search strategy from the main application by an interface. Generally, if you have a hard-coded implementation, and you realize that you need to change it in a significant way, that’s a good opportunity to extract it and hide it behind an interface.

A good interface choice is a bit tricky here, however. In the original implementation, we could say, “process the batch of numbers (search for counterexamples) between 1 and 2 million.” When that batch is saved to the database, we would start on the next batch, and all the batches would be the same size, so (ignoring that computing \sigma(n) the old way takes longer as n grows) each batch required roughly the same time to run.

The new search strategy doesn’t have a sensible way to do this. You can’t say “start processing from K” because we don’t know how to easily get from K to the parameter of the enumeration corresponding to K (if one exists). This is partly because our enumeration isn’t monotonic increasing (2^1 3^1 5^1 = 30 comes before 2^4 = 16). And partly because even if we did have a scheme, it would almost certainly require us to compute a prime factorization, which is slow. It would be better if we could save the data from the latest step of the enumeration, and load it up when starting the next batch of the search.

This scheme suggests a nicely generic interface for stopping and restarting a search from a particular spot. The definition of a “spot,” and how to start searching from that spot, are what’s hidden by the interface. Here’s a first pass.

SearchState = TypeVar('SearchState')

class SearchStrategy(ABC):
    @abstractmethod
    def starting_from(self, search_state: SearchState) -> SearchStrategy:
        '''Reset the search strategy to search from a given state.'''
        pass

    @abstractmethod
    def search_state(self) -> SearchState:
        '''Get an object describing the current state of the enumeration.'''
        pass

    @abstractmethod
    def next_batch(self, batch_size: int) -> List[RiemannDivisorSum]:
        '''Process the next batch of Riemann Divisor Sums'''
        pass

Note that SearchState is defined as a generic type variable because we cannot say anything about its structure yet. The implementation class is responsible for defining what constitutes a search state, and getting the search strategy back to the correct step of the enumeration given the search state as input. Later I realized we do need some structure on the SearchState—the ability to serialize it for storage in the database—so we elevated it to an interface later.

Also note that we are making SearchStrategy own the job of computing the Riemann divisor sums. This is because the enumeration details and the algorithm to compute the divisor sums are now coupled. For the exhaustive search strategy it was “integers n, naively loop over smaller divisors.” In the new strategy it’s “prime factorizations, prime-factorization-based divisor sum.” We could decouple this, but there is little reason to now because the implementations are still in 1-1 correspondence.

This commit implements the old search strategy in terms of this interface, and this commit implements the new search strategy. In the latter, I use pytest.parameterize to test against the interface and parameterize over the implementations.

The last needed bit is the ability to store and recover the search state in between executions of the main program. This requires a second database table. The minimal thing we could do is just store and update a single row for each search strategy, providing the search state as of the last time the program was run and stopped. This would do, but in my opinion an append-only log is a better design for such a table. That is, each batch computed will have a record containing the timestamp the batch started and finished, along with the starting and ending search state. We can use the largest timestamp for a given search strategy to pick up where we left off across program runs.

One can imagine this being the basis for an application like folding@home or the BOINC family of projects, where a database stores chunks of a larger computation (ranges of a search space), clients can request chunks to complete, and they are assembled into a complete database. In this case we might want to associate the chunk metadata with the computed results (say, via a foreign key). That would require a bit of work from what we have now, but note that the interfaces would remain reusable for this. For now, we will just incorporate the basic table approach. It is completed in this pull request, and tying it into the main search routine is done in this commit.

However, when running it with the superabundant search strategy, we immediately run into a problem. Superabundant numbers grow too fast, and within a few small batches of size 100 we quickly exceed the 64 bits available to numba and sqlite to store the relevant data.

>>> fac = partition_to_prime_factorization(partitions_of_n(16)[167])
>>> fac2 = [p**d for (p, d) in fac]
>>> fac2
[16, 81, 625, 2401, 11, 13, 17, 19, 23, 29, 31, 37]
>>> math.log2(reduce(lambda x,y: x*y, fac2))
65.89743638933722

Running populate_database.py results in the error

$ python -m riemann.populate_database db.sqlite3 SuperabundantSearchStrategy 100
Searching with strategy SuperabundantSearchStrategy
Starting from search state SuperabundantEnumerationIndex(level=1, index_in_level=0)
Computed [1,0, 10,4] in 0:00:03.618798
Computed [10,4, 12,6] in 0:00:00.031451
Computed [12,6, 13,29] in 0:00:00.031518
Computed [13,29, 14,28] in 0:00:00.041464
Computed [14,28, 14,128] in 0:00:00.041674
Computed [14,128, 15,93] in 0:00:00.034419
...
OverflowError: Python int too large to convert to SQLite INTEGER

We’ll see what we can do about this in a future article, but meanwhile we do get some additional divisor sums for these large numbers, and 10080 is still the best.

sqlite> select n, witness_value 
from RiemannDivisorSums 
where witness_value > 1.7 and n > 5040 
order by witness_value desc 
limit 10;

10080|1.7558143389253
55440|1.75124651488749
27720|1.74253672381383
7560|1.73991651920276
15120|1.73855867428903
160626866400|1.73744669257158
321253732800|1.73706925385011
110880|1.73484901030336
6983776800|1.73417642212953
720720|1.73306535623807

Searching for RH Counterexamples — Adding a Database

In the last article we set up pytest for a simple application that computes divisor sums \sigma(n) and tries to disprove the Riemann Hypothesis. In this post we’ll show how to extend the application as we add a database dependency. The database stores the computed sums so we can analyze them after our application finishes.

As in the previous post, I’ll link to specific git commits in the final code repository to show how the project evolves. You can browse or checkout the repository at each commit to see how it works.

Interface before implementation

The approach we’ll take is one that highlights the principle of good testing and good software design: separate components by thin interfaces so that the implementations of those interfaces can change later without needing to update lots of client code.

We’ll take this to the extreme by implementing and testing the logic for our application before we ever decide what sort of database we plan to use! In other words, the choice of database will be our last choice, making it inherently flexible to change. That is, first we iron out a minimal interface that our application needs, and then choose the right database based on those needs. This is useful because software engineers often don’t understand how the choice of a dependency (especially a database dependency) will work out long term, particularly as a prototype starts to scale and hit application-specific bottlenecks. Couple this with the industry’s trend of chasing hot new fads, and eventually you realize no choice is sacred. Interface separation is the software engineer’s only defense, and their most potent tool for flexibility. As a side note, Tom Gamon summarizes this attitude well in a recent article, borrowing the analogy from a 1975 investment essay The Winner’s Game by Charles Ellis. Some of his other articles reinforce the idea that important decisions should be made as late as possible, since that is the only time you know enough to make those decisions well.

Our application has two parts so far: adding new divisor sums to the database, and loading divisor sums for analysis. Since we’ll be adding to this database over time, it may also be prudent to summarize the contents of the database, e.g. to say what’s the largest computed integer. This suggests the following first-pass interface, implemented in this commit.

class DivisorDb(ABC):
    @abstractmethod
    def load() -> List[RiemannDivisorSum]:
        '''Load the entire database.'''
        pass

    @abstractmethod
    def upsert(data: List[RiemannDivisorSum]) -> None:
        '''Insert or update data.'''
        pass

    @abstractmethod
    def summarize() -> SummaryStats:
        '''Summarize the contents of the database.'''
        pass

RiemannDivisorSum and SummaryStats are dataclasses. These are special classes that are intended to have restricted behavior: storing data and providing simple derivations on that data. For us this provides a stabler interface because the contents of the return values can change over time without interrupting other code. For example, we might want to eventually store the set of divisors alongside their sum. Compare this to returning a list or tuple, which is brittle when used with things like tuple assignment.

The other interesting tidbit about the commit is the use of abstract base classes (“ABC”, an awful name choice). Python has limited support for declaring an “interface” as many other languages do. The pythonic convention was always to use its “duck-typing” feature, which meant to just call whatever methods you want on an object, and then any object that supports has those methods can be used in that spot. The mantra was, “if it walks like a duck and talks like a duck, then it’s a duck.” However, there was no way to say “a duck is any object that has a waddle and quack method, and those are the only allowed duck functions.” As a result, I often saw folks tie their code to one particular duck implementation. That said, there were some mildly cumbersome third party libraries that enabled interface declarations. Better, recent versions of Python introduced the abstract base class as a means to enforce interfaces, and structural subtyping (typing.Protocol) to interact with type hints when subtyping directly is not feasible (e.g., when the source is in different codebases).

Moving on, we can implement an in-memory database that can be used for testing. This is done in this commit. One crucial aspect of these tests is that they do not rely on the knowledge that the in-memory database is secretly a dictionary. That is, the tests use only the DivisorDb interface and never inspect the underlying dict. This allows the same tests to run against all implementations, e.g., using pytest.parameterize. Also note it’s not thread safe or atomic, but for us this doesn’t really matter.

Injecting the Interface

With our first-pass database interface and implementation, we can write the part of the application that populates the database with data. A simple serial algorithm that computes divisor sums in batches of 100k until the user hits Ctrl-C is done in this commit.

def populate_db(db: DivisorDb, batch_size: int = 100000) -> None:
    '''Populate the db in batches.'''
    starting_n = (db.summarize().largest_computed_n or 5040) + 1
    while True:
        ending_n = starting_n + batch_size
        db.upsert(compute_riemann_divisor_sums(starting_n, ending_n))
        starting_n = ending_n + 1

I only tested this code manually. The reason is that line 13 (highlighted in the abridged snippet above) is the only significant behavior not already covered by the InMemoryDivisorDb tests. (Of course, that line had a bug later fixed in this commit). I’m also expecting to change it soon, and spending time testing vs implementing features is a tradeoff that should not always fall on the side of testing.

Next let’s swap in a SQL database. We’ll add sqlite3, which comes prepackaged with python, so needs no dependency management. The implementation in this commit uses the same interface as the in-memory database, but the implementation is full of SQL queries. With this, we can upgrade our tests to run identically on both implementations. The commit looks large, but really I just indented all the existing tests, and added the pytest parameterize annotation to the class definition (and corresponding method arguments). This avoids adding a parameterize annotation to every individual test function—which wouldn’t be all that bad, but each new test would require the writer to remember to include the annotation, and this way systematically requires the extra method argument.

And finally, we can switch the database population script to use the SQL database implementation. This is done in this commit. Notice how simple it is, and how it doesn’t require any extra testing.

After running it a few times and getting a database with about 20 million rows, we can apply the simplest possible analysis: showing the top few witness values.

sqlite> select n, witness_value from RiemannDivisorSums where witness_value > 1.7 order by witness_value desc limit 100;
10080|1.7558143389253
55440|1.75124651488749
27720|1.74253672381383
7560|1.73991651920276
15120|1.73855867428903
110880|1.73484901030336
720720|1.73306535623807
1441440|1.72774021157846
166320|1.7269287425473
2162160|1.72557022852613
4324320|1.72354665986337
65520|1.71788900114772
3603600|1.71646721405987
332640|1.71609697536058
10810800|1.71607328780293
7207200|1.71577914933961
30240|1.71395368739173
20160|1.71381061514181
25200|1.71248203640096
83160|1.71210965310318
360360|1.71187211014506
277200|1.71124375582698
2882880|1.7106690212765
12252240|1.70971873843453
12600|1.70953565488377
8648640|1.70941081706371
32760|1.708296575835
221760|1.70824623791406
14414400|1.70288499724944
131040|1.70269370474016
554400|1.70259313608473
1081080|1.70080265951221

We can also confirm John’s claim that “the winners are all multiples of 2520,” as the best non-multiple-of-2520 up to 20 million is 18480, whose witness value is only about 1.69.

This multiple-of-2520 pattern is probably because 2520 is a highly composite number, i.e., it has more divisors than all smaller numbers, so its sum-of-divisors will tend to be large. Digging in a bit further, it seems the smallest counterexample, if it exists, is necessarily a superabundant number. Such numbers have a nice structure described here that suggests a search strategy better than trying every number.

Next time, we can introduce the concept of a search strategy as a new component to the application, and experiment with different search strategies. Other paths forward include building a front-end component, and deploying the system on a server so that the database can be populated continuously.