Increasingly capable Artificial Intelligence (AI) technologies are gaining steam in proving long-standing mathematical conjectures. The latest development involves a proof of another one of prolific mathematician Paul Erdős' 1,135 unsolved problems, but unlike AI's previous accomplishments in tackling part of Erdős' legacy, GPT-5.4's proof of Erdős #1196 appears to be genuinely novel.

The previous proofs by AI systems of other Erdős problems did not answer the question of whether the technology was bringing anything new to the table. The problems themselves could be considered "low-hanging fruit", whose unsolved status had more to do with the obscurity of the conjectures within the Erdős collection than their difficulty. When AI developed proofs for them, it more or less followed the playbook that mathematicians had established in proving other Erdős conjectures.

But Erdős' 1196th conjecture is not in that category. Mathematicians had previously taken on the challenge of developing a proof for it, largely turning to the tools of probability and statistics in the process. But instead of copying that approach, GPT-5.4 Pro found the path for proving the conjecture differently. Mathematician Terrence Tao offered this observation:

I had previously stated the opinion that the AI-generated proof had inadvertently highlighted a tighter connection between the anatomy of integers and the theory of Markov chains than had previously been explicitly noted in the literature. Based on further developments, I would like to update that opinion to the following: the AI-generated proof artefact, when combined with subsequent (and mostly human-generated) analysis, has revealed a tight connection between the anatomy of integers and flow network theory that does not, to my knowledge, have any explicit precursor in the literature (although related uses of Markov chains in adjacent settings do appear in that literature).

The development of the proof was verified using the Lean proof assistant, which we would argue is the secret sauce behind why AI technologies are making such rapid progress in advancing proofs to unsolved problems in the field. The pairing of the technologies is key to the advancement.

Getting back to the novelty of the proof, here's a comment by Jared Duker Lichtman, who developed the first proof of the related Erdős primitive set conjecture as part of his doctoral thesis in 2024.

In my doctorate, I proved the Erdős Primitive Set Conjecture, showing that the primes themselves are maximal among all primitive sets.

This problem will always be in my heart: I worked on it for 4 years (even when my mentors recommended against it!) and loved every minute of it.

[Primitive sets are a vast generalization of the prime numbers: A set S is called primitive if no number in S divides another.]

Now Erdős#1196 is an asymptotic version of Erdős' conjecture, for primitive sets of "large" numbers. It was posed in 1966 by the Hungarian legends Paul Erdős, András Sárközy, and Endre Szemerédi.

I'd been working on it for many years, and consulted/badgered many experts about it, including my mentors Carl Pomerance and James Maynard.

The proof produced by GPT5.4 Pro was quite surprising, since it rejected the "gambit" that was implicit in all works on the subject since Erdős' original 1935 paper. The idea to pass from analysis to probability was so natural & tempting from a human-conceptual point of view, that it obscured a technical possibility to retain (efficient, yet counter-intuitve) analytic terminology throughout, by use of the von Mangoldt function \Lambda(n).

The closest analogy I would give would be that the main openings in chess were well-studied, but AI discovers a new opening line that had been overlooked based on human aesthetics and convention.

In fact, the von Mangoldt function itself is celebrated for it's connection to primes and the Riemann zeta function--but its piecewise definition appears to be odd and unmotivated to students seeing it for the first time. By the same token, in Erdős#1196, the von Mangoldt weights seem odd and unmotivated but turn out to cleverly encode a fundamental identity \sum_{q|n}\Lambda(q) = \log n, which is equivalent to unique factorization of n into primes. This is the exact trick that breaks the analytic issues arising in the "usual opening".

Joshua Zelinsky offers perhaps the best framing of the accomplishment and what it could mean for additional progress:

Four things to note: #1196 is a decently well known problem. It wasn’t like Erdős-Straus level fame, but it is well known enough that I was familiar with it. Second, this is not a problem where no one had worked on it; there was a lot of prior work on it and closely related problems. Third, this is not example where the AI made small modifications to things in the literature or recognized that large parts of the problem were in an obscure paper. The approach the AI used is largely a different direction than the literature on this problem went. Fourth, and closely related to three, this proof does look like parts of it will inspire subsequent proofs because it really is going in a different direction which now looks likely to be a productive line of investigation for similar problems.

If this kind of progress continues, the productivity of AI technology in cracking unsolved math problems will be the biggest math story of the year. AI was already the biggest math story of 2025, but 2026 is shaping up to be even more so.

The rolling twelve-month average price of a 10.75 fluid ounce can of Campbell's Condensed Tomato Soup continued falling in the three months since our last monthly price history update.

In that January 2026 snapshot, the rolling average was $1.21 per can, which was down from an average of $1.29 per can in September 2025. In mid-April 2026, that twelve-month average has fallen further, dropping to $1.15 per can.

The falling prices during this period, which coincides with peak "soup season" in the United States, also coincide with deep discounts for consumers at major grocery selling retailers. Here are the prices available to U.S. consumers in April 2026 at ten of the nation's largest grocery-selling retailers and how they've changed since our January 2026 snapshot:

• Walmart: $1.00/each, decrease of $0.07 (-6.5%)
• Amazon: $1.00/each, unchanged
• Kroger: $1.00/each, unchanged
• Walgreens: $1.99/each, unchanged
• Target: $1.29/each, decrease of $0.10 (-7.2%)
• CVS: $2.49/each, unchanged
• Albertsons: $1.29/each, increase of $0.20 (+15.5%)
• Food Lion: $1.00/each, decrease of $0.25 (-20.0%)
• H-E-B: $1.24/each, unchanged
• Meijer: $1.29/each, unchanged

Not all these prices carry equal weight. It would not surprise us to learn that Walmart is the largest single purveyor of Campbell's tomato soup in the U.S., followed by Kroger-affiliated grocery stores. The dominance in sales of these "Big Two" gives more weight to their prices as being more representative of what most Americans pay on average for a can of Campbell's tomato soup.

The falling rolling average price of Campbell's tomato soup is the result of substantial discounts at these major retailers since September 2025. During this period, an iconic 10.75 fluid ounce "Number 1" size can of Campbell's Condensed Tomato Soup has been priced at just $1.00 per can at these biggest sellers.

The following chart presents the price history of Campbell's Condensed Tomato Soup in the 21st century, from January 2000 through April 2026. If you want to see more history, you can find our collected price data extending back to January 1898 here:

Before 2022, it was rare to ever find a can of Campbell's tomato soup priced higher than $1.00 per can. However, because of the high inflation unleashed by the Biden administration, it has been rare since 2022 to find a Campbell's tomato soup on sale for less than $1.00 per can.

But will this sale price hold now that soup season is ending? We'll check back in on the latest prices for our next update in three months.

Image Credit: Campbell's Tomato Soup at Walmart on 20 February 2026 with digital shelf price tags photo by Iron Man on Unsplash.

The climbing limo method of forecasting future GDP in the United States projects the nation's economic output in the recently finished first quarter of 2026 will be around $31.5 trillion.

This estimate assumes the momentum the U.S. economy recorded in growing between 2024-Q4 and 2025-Q2 will be sustained through the recently ended quarter of 2026-Q1. Since the U.S. economy's momentum has come in stronger than expected over the intervening quarters since our last snapshot, it's likely 2026-Q1's will come in above that value, which is not adjusted for inflation.

These projections have been delayed because the Senate Democrats' government shutdown disrupted U.S. economic data reporting. It took until earlier this month to get finalized GDP data for both 2025-Q3 and Q4, which means we can now generate the climbing limo method's momentum-based projections for both 2026-Q2 and 2026-Q3. The following chart visually presents those projections while showing how actual non-inflation adjusted GDP tracked with the model's previous projections:

For the now current quarter of 2026-Q2, the climbing limo GDP forecasting method projects GDP will potentially rise to almost $32.8 trillion. Since that projection was generated with finalized GDP figures for 2025-Q1 and 2025-Q3, long before any impact from the Iran war would be felt, it will be interesting to see how well that forecast tracks with 2026-Q2's actual GDP.

That's because the climbing limo forecasting method is a momentum-based projection. As such, even when recorded GDP deviates considerably from the forecast values that are projected three quarters ahead in time, it provides valuable information in confirming the economy's underlying momentum has changed. We should get a good reading on how big the Iran war's impact has been on the U.S. economy after the actual GDP data for 2026-Q2 is reported.

Meanwhile, the most distant future projection we can make with available finalized GDP data is for 2026-Q3, where the climbing limo forecasting method anticipates the nation's nominal GDP will rise to about $32.9 trillion.

References

U.S. Bureau of Economic Analysis. National Income and Product Accounts. Table 1.1.5. Gross Domestic Product. [Online Database]. Accessed 9 April 2026.

Image Credit: Microsoft Copilot Designer. Prompt: "A simple sketch of a limousine driving uphill toward the right side of a rising zig zag line chart".

Measured by the number of companies announcing they would be lowering their dividend payouts from their previous level, March 2026 was a good month for the dividend paying companies of the U.S. stock market. But then, after the concentrated carnage that swept through the ranks of publicly traded Business Development Companies in February 2026, there aren't many BDCs left in the financial services arena with imminent dividend cuts in their future.

That's not to say there won't be more such firms that will need to face up to the risk of their loans to software firms now facing pressure from AI technologies going bad, but the month of March saw considerably fewer of them.

Overall, the number of dividend decreasing companies in our sampling for the month dropped from a revised total of 27 the month before to just 6 in March 2026. The following chart presents the tally of dividend decreases recorded for each industrial sector within the U.S. stock market.

The oil and gas sector saw the most decreases with four firms announcing decreased dividends in March 2026. Three of which are represented by royalty trusts that pay variable dividends, which directly results from the business conditions for this portion of the industry in the preceding month. While March 2026 saw oil and gas prices surge in response to the Iran war's disruption of the global trade in oil, February had seen falling prices, which reduced the revenue and earnings of many of these firms, which in turn resulted in their reduced distributions to their shareholders for this month.

Even so, the number of these firms falls well below the threshold of ten we use as a rule of thumb for determining whether the industry is facing contractionary conditions.

The financial services sector added two more dividend decreasing firms to the total, both of which pay fixed dividends that required action by their boards of directors to declare.

Here are the sampled dividend decreases for March 2026:

• Monroe Capital (Mortgage) (NASDAQ: MRCC)
• FS Credit Opportunities (NYSE: FSCO)
• Cross Timbers Royalty Trust (NYSE: CRT)
• Permian Basin Royalty Trust (NYSE: PBT)
• PermRock Royalty Trust (NYSE: PRT)
• FutureFuel (NYSE: FF)

Monroe Capital has a strange situation in that after the company slashed its dividend payout in March, it delivered an outsize dividend payment in April 2026 after its acquisition by Horizon Technology Finance (NASDAQ: HRZN).

The overall number of dividend decreases announced in March 2026 is well below the total of 50 that signifies recessionary conditions are present in the U.S. economy as a whole.

Image credit: Dividends definition. State Savings and Loan Association advertisement on Page 9 of Beatrice, Nebraska's Beatrice Daily Express, 12 December 1921. Chronicling America. [Online Database]. 12 December 1921. Public domain image.

The S&P 500 (Index: SPX) continued its bullish run in the trading week ending on Friday, 17 April 2026. The index closed out the week at 7,126.06, a new record high following 13 consecutive days of higher closes.

To put that unlikely event into perspective, since 3 January 1950, the S&P 500 has only seen one winning streak that has exceeded 12 consecutive days. That streak started on 26 March 1971 and ended on 15 April 1971. The index rose from 99.61 at the close of trading on 25 March 1971 to reach 103.52 after 14 trading days, a gain of 3.9%.

The current winning streak, which is still running going into the new trading week, has seen the S&P 500 rise 12.3% after the index bottomed at 6,343.72 on 30 March 2026. The gains have largely come as the U.S. has largely achieved its aims in the Iran war, including an announcement from Iran's government that shipping traffic would resume transiting the Hormuz Strait without harassment on Friday, 17 April 2026.

Investors responded positively, bidding oil prices substantially lower which, in turn, boosted stock prices. The latest update of the alternative futures chart finds the index' trajectory has risen to fall within the redzone forecast range we added several weeks ago before the geopolitical conflict, where it is now just 2.6% below the central trend line of the forecast range.

The positive development also improved the outlook for a Federal Reserve rate cut in 2026, which is a big change after the last few weeks had all but taken that chance off the table. The CME Group's FedWatch Tool projects a rising chance of a single quarter point rate cut in 2026, most likely to be announced after the Fed's Open Market Committee meets on 9 December (2026-Q4).

Of course, whether that happens depends on the course of geopolitical events in the Iran war, which the flow of new information in recent weeks has demonstrated to be subject to change with little notice. To get a sense of what we mean, here are the market moving headlines from just the past week:

Monday, 13 April 2026

• Signs and portents for the U.S. economy:
• Vance: 'No agreement reached' in US and Iran peace talks
• Oil prices jump 4% on US blockade of Iran after peace talks collapse
• What does a US naval blockade of Iran mean for oil flows?
• US existing home sales drop to nine-month low in March amid tight supply
• Mixed growth signs, bigger trouble developing in China:
• China poised for Q1 GDP growth rebound but Iran war dims 2026 outlook: Reuters poll
• China's exports set to lose momentum as Iran war undercuts AI-driven boom: Reuters poll
• China March new loans jumps less than expected, no sign of fresh policy easing
• China court orders bankruptcy liquidation for Zhongzhi Group and 316 affiliates
• China Starts Bankruptcy Liquidation of Shadow Bank Zhongzhi
• BOJ minions starting to think that boosting value of Japan's currency might keep inflation at bay:
• BOJ policy to boost yen could be an option to curb inflation, Japanese minister says
• BOJ highlights market volatility, economic hit from Middle East conflict
• Bigger trouble developing in Eurozone:
• Physical oil in Europe hits record high near $150 a barrel as Hormuz crisis worsens
• ECB minions say Eurozone energy prices are right where they planned, not as sure about whether they'll hike interest rates as the market:
• ECB's Vujcic says energy prices are still closest to ECB basic scenario
• ECB rate rise will depend on effects of oil price surge, De Guindos says
• Euro zone bond yields rise as markets up ECB rate hike bets
• Wall Street rallied and closed higher despite failed U.S.-Iran talks
• Wall Street indexes gain as investors hold out hope for US-Iran resolution

Tuesday, 14 April 2026

• Signs and portents for the U.S. economy:
• US producer inflation rises below expectations in March, but war fires up energy prices
• Jet fuel shock from Iran war worsens crisis for global airlines
• Crude oil plunges on potential for renewed U.S.-Iran talks, pressure from naval blockade
• Oil prices fall for a second day on expectations US-Iran talks may resume
• Fed minions continue pushing back on possibility of rate cuts before 2027:
• Fed's Goolsbee says rate cuts may need to wait until 2027
• Bigger trouble, stimulus developing in China:
• China's export engine stutters as Iran war chills global demand
• China copper smelters may press ahead with production curbs amid acid export ban
• Exclusive: China finance ministry to meet underwriters on special treasury bond issuance plan
• China's March crude oil imports unscathed by Iran war, gas imports hit a multi-year low
• US Treasury's Bessent says China has been unreliable partner by hoarding oil during war
• ECB minions dithering as bigger trouble developing in Eurozone:
• ECB has not made its mind up on rates, Lagarde says
• ECB rate hike not self-evident, policymaker Rehn says
• Euro zone faces big growth hit even if Iran war quickly resolved, IMF says
• Wall Street rallies on renewed hopes for US-Iran talks, earnings boost
• S&P 500 jumps and these 20 stocks are fueling the rally
• Higher oil prices, higher yields, no more rate cuts? No problem for US stocks
• U.S. markets continue to be largely isolated from the Iran war oil shock – strategist

Wednesday, 15 April 2026

• Signs and portents for the U.S. economy:
• Oil prices fall as hopes for US-Iran deal outweigh supply disruption concerns
• Economic growth will be slower this quarter amid war, US Treasury chief says
• Treasury chief says US growth may exceed 3% or 3.5% this year despite Iran war
• US import prices increase below expectations; sharp rise anticipated due to Iran war
• Former Fed minion tries to put 2026 rate cut back on table, current Fed minions say inflation likely to stay near the Fed's real target and interest rates likely to stay where they're at as chief minion pressured to resign:
• Former US Treasury Secretary Yellen says one Fed rate cut possible this year
• Exclusive: Fed's Musalem says oil shock likely to keep core inflation near 3%, rates on hold for some time
• Fed's Hammack: Rates likely on hold 'for a good while' - CNBC
• Trump threatens to fire Powell if he doesn't quit Fed board; ongoing probe clouds Warsh confirmation
• BOJ minions face a growing lack of support for their rate hike plans:
• More Japanese firms say no to rate hikes as Iran war clouds outlook: Reuters poll
• ECB minions say they're doing great with Eurozone monetary policies:
• ECB in 'relatively favorable' policy position, Schnabel says
• ECB policymakers see little evidence for now for April hike, sources say
• S&P 500 closes at fresh record, recovering all losses since start of US-Iran war
• S&P 500, Nasdaq close at record as investors price in positive Middle East developments

Thursday, 16 April 2026

• Signs and portents for the U.S. economy:
• Oil prices rise on doubts US-Iran peace talks will ease Hormuz disruption
• Gas prices are dropping. These states are seeing the biggest relief
• Iran war brings US close to net crude exporter for first time since World War Two
• Fed minions say less likely to cut rates in months ahead:
• Fed's Miran may scale back rate-cut outlook again due to 'less favorable' inflation developments
• Fed's Williams says war is already driving up inflation pressures
• Growth signs, continuing trouble developing in China:
• China's Q1 GDP grows 5.0% y/y, tops market forecast
• China's new home prices extend decline despite improvement in major cities
• BOJ minions expected to hike Japan's interest rates before summer:
• BOJ to hike rates by June as war-fuelled inflation risks mount: Reuters poll
• ECB minions say they're calm for now, not excited enough to hike Eurozone interest rates:
• Calm judgment will prevail over haste at ECB, Rehn says
• ECB policymakers play down April rate hike chances
• Wall Street enjoys second day of fresh highs as Trump announces Middle East peace advancements
• S&P 500, Nasdaq edge up to new records with Middle East hopes in focus

Friday, 17 April 2026

• Signs and portents for the U.S. economy:
• Iran agrees to suspend nuclear program, Trump says - report
• Oil prices plunge 12% after Iran says Strait of Hormuz is open for commercial vessels
• US buyers redirect imported fertilizer overseas as Iran war drives up global prices
• Fed minions thinking rate cuts in 2026 might not yet be dead:
• Fed's Waller says swift war end could keep rate cut hopes alive
• Fed chair nominee Warsh will have ideas, economy may deliver surprises, Daly says
• No stimulus developing in China:
• China expected to keep benchmark lending rates steady after strong GDP data
• Analysts doubt Iran war will prompt China stimulus, tip ‘solid’ first-quarter GDP growth
• BOJ minions getting excited to hike Japan's interest rates again, but maybe not in April 2026:
• BOJ's hawkish hints keep rate hike on the cards
• Japan's core inflation likely ticked up in March fuelled by energy costs: Reuters poll
• BOJ chief avoids hints of April rate hike, shattering hawkish market bets
• Chief ECB minion starting to think Iran war might somehow affect Eurozone economy:
• Iran war presents risk to euro zone growth and inflation outlook, Lagarde says
• IMF expects ECB to raise rates by half a percentage point in 2026
• S&P 500 hits record high after Iran declares Strait of Hormuz open
• Wall Street indexes hit record highs as oil falls with Strait of Hormuz declared open
• S&P 500’s bottom to record levels marks fastest recovery in 36 years

We normally cut off our coverage of a trading week's news events with the close of trading each week. Since that's the case, be aware that events that come up afterward can have a material effect on how stock futures trade in the interim between the previous week's market close and the upcoming week's market open.

In other news, the Atlanta Fed's GDPNow tool forecast of real GDP growth in 2026-Q1 was unchanged at +1.3%, the same as at the end of the preceding week.

Image credit: Microsoft Copilot Designer. Prompt: "An editorial cartoon of a suit-wearing Wall Street bull and bear celebrating falling oil prices because the Hormuz Strait is open. The bull holds a newspaper with the headline 'HORMUZ STRAIT OPEN! OIL PRICES PLUNGE!' and a champagne glass, while the bear laughs holding dollar bills. A toppled oil barrel and a downward oil price chart sit on the table."