For relatively prime positive integers ${{u}_{0}}$ and $r$ , we consider the least common multiple ${{L}_{n}}\,:=\,\text{lcm}\left( {{u}_{0}},\,{{u}_{1}},\,.\,.\,.\,,\,{{u}_{n}} \right)$ of the finite arithmetic progression $\left\{ {{u}_{k}}\,:=\,{{u}_{0}}\,+\,kr \right\}_{k=0}^{n}$ . We derive new lower bounds on ${{L}_{n}}$ that improve upon those obtained previously when either ${{u}_{0}}$ or $n$ is large. When $r$ is prime, our best bound is sharp up to a factor of $n\,+\,1$ for ${{u}_{0}}$ properly chosen, and is also nearly sharp as $n\,\to \,\infty$ .

Greenhouse and laboratory studies were conducted to confirm and quantify glyphosate resistance, quantify pyrithiobac resistance, and investigate interaction between flumiclorac and glyphosate mixtures on control of Palmer amaranth from Mississippi. The GR50 (herbicide dose required to cause a 50% reduction in plant growth) values for two glyphosate-resistant biotypes, C1B1 and T4B1, and a glyphosate-susceptible (GS) biotype were 1.52, 1.3, and 0.09 kg ae ha−1 glyphosate, respectively. This indicated that the C1B1 and T4B1 biotypes were 17- and 14-fold resistant to glyphosate, respectively, compared with the GS biotype. The C1B1 and T4B1 biotypes were also resistant to pyrithiobac, an acetolactate synthase (ALS) inhibitor, with GR50 values of 0.06 and 0.07 kg ai ha−1, respectively. This indicated that the C1B1 and T4B1 biotypes were 7- and 8-fold, respectively, more resistant to pyrithiobac compared with the GS biotype, which had a GR50 value of 0.009 kg ha−1. Flumiclorac was antagonistic to glyphosate by reducing glyphosate translocation. The C1B1 and T4B1 absorbed less glyphosate 48 h after treatment (HAT) compared with the GS biotype. The majority of the translocated glyphosate accumulated in the shoot above the treated leaf (that contains the apical meristem) in the GS biotype and in the shoot below the treated leaf in the resistant biotypes, C1B1 and T4B1, by 48 HAT. The C1B1 biotype accumulated negligible shikimate levels, whereas the T4B1 and GS biotypes recorded elevated levels of shikimate. Metabolism of glyphosate to aminomethylphosphonic acid was not detected in either of the resistant biotypes or the susceptible GS biotype. The above results confirm multiple resistance to glyphosate and pyrithiobac in Palmer amaranth biotypes from Mississippi and indicate that resistance to glyphosate is partly due to reduced absorption and translocation of glyphosate.

A threefold glyphosate tolerance was identified in two Italian ryegrass populations, T1 and T2, from Mississippi. Laboratory experiments were conducted to characterize the mechanism of glyphosate tolerance in these populations. The T1 population absorbed less 14C-glyphosate (43% of applied) compared to the susceptible (S) population (59% of applied) at 48 h after treatment (HAT). The T2 population absorbed 14C-glyphosate at levels (56% of applied at 48 HAT) that were similar to both T1 and S populations, but tended to be more comparable to the S population. The amount of 14C-glyphosate that remained in the treated leaf was significantly higher in both T1 (67% of absorbed) and T2 (65% of absorbed) populations compared to the S population (45% of absorbed) at 48 HAT. The amount of 14C-glyphosate that moved out of treated leaf to shoot and root was lower in both T1 (25% of absorbed in shoot and 9% of absorbed in root) and T2 (25% of absorbed in shoot and 11% of absorbed in root) populations compared to the S population (40% of absorbed in shoot and 16% of absorbed in root) at 48 HAT. There were no differences in epicuticular wax mass among the three populations. Treating a single leaf with glyphosate solution at the field use rate (0.84 kg ae ha−1) as 10 1-µl droplets killed the S plant but not the T1 and T2 plants (33 and 55% shoot fresh-weight reduction, respectively). Shikimic acid accumulated rapidly at higher levels in glyphosate-treated leaf segments of the S population compared to the T1 population up to 100 µM glyphosate. However, above 500 µM glyphosate, the levels of shikimate were similar in both the S and T1 populations. Furthermore, shikimic acid content was three- to sixfold more in whole plants of the S population treated with 0.22 kg ae ha−1 glyphosate compared to the T1 and T2 populations. No degradation of glyphosate to aminomethylphosphonic acid was detected among the tolerant and susceptible populations. These results indicate that tolerance to glyphosate in the T1 population is partly due to reduced absorption and translocation of glyphosate and in the T2 population it is partly due to reduced translocation of glyphosate.